CHAPTER
7
Multiplying and Dividing
Polynomials
GET READY
362
Math Link
364
7.1 Warm Up
366
7.1 Multiplying and Dividing Monomials
367
7.2 Warm Up
378
7.2 Multiplying Polynomials by Monomials
379
7.3 Warm Up
387
7.3 Dividing Polynomials by Monomials
388
Graphic Organizer
396
Chapter 7 Review
397
Key Word Builder
401
Chapter 7 Practice Test
402
Math Link: Wrap It Up!
405
Challenge
406
Chapters 5 –7 Review
407
Task
412
Answers
414
Name: _____________________________________________________
Date: ______________
Get Ready
Language of Polynomials
term
● a number or variable or the product of a number and a variable
● examples: 5, x, 3y
polynomial
● an expression made up of 1 or more terms connected by addition or subtraction
monomial
2
● a polynomial with 1 term. Examples: 4x, 5, 3a
binomial
2
● a polynomial with 2 terms. Examples: 4x + 5, 3a – 2a
trinomial
2
● a polynomial with 3 terms. Example: 4x – 8x + 2
To find the degree of a term, add the exponents of the variable(s) in the term.
Example: 5x2y = 5x2y1, so it has a degree of 3.
The degree of a polynomial is the degree of the term with the highest degree.
Example: 7b2 + 3b – 11 has a degree of 2 because the highest degree term, 7b2, has a degree of 2.
1.
monomial, binomial, or trinomial
Complete the table.
Type of Polynomial
Degree of Polynomial
a) x 2 – 2x + 5
b) 11c + 14
c) 24d 2
Equivalent Expressions
like terms have the same variable(s) with the
same exponent(s)
combine by adding or subtracting
examples: 5x – 3x = 2x, − 4k 2 + 0.5k 2 = –3.5k 2
●
●
●
2.
●
●
●
unlike terms have different variables or
like variables with different exponents
cannot be combined
examples: 2t + 2t 2, −pq + 6p
Simplify by combining like terms.
x 2 – 6x + 2x 2 + 5x
a)
b) –2g2 – 7n + 4g2 + 3n
= x 2 – 6x + 2x 2 + 5x
= x 2 + 2x 2 – 6x + 5x
=
362
x2 – x
Group like terms.
Combine like terms.
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
Date: ______________
Using a Model to Add and Subtract Polynomials
You can model adding or subtracting polynomials to simplify the expression.
(3x2 + 2x)
+
(x2 – x + 2)
= positive x2-tile
= positive x-tile
Group like terms. Then, remove any zero pairs.
= negative x-tile
= positive 1-tile
4x2 + x + 2
3.
Draw a model to add or subtract the polynomials.
a) (2x 2 – 3x + 1) + (4x – 5)
b) (3x 2 + 2x + 1) – (x 2 + x)
Using Opposites to Subtract Polynomials
To find the opposite of a polynomial, write the opposite of each term.
Example: The opposite of 2x2 + 3x – 7 is –2x2 – 3x + 7.
To subtract polynomials, add the opposite.
(4x2 + x + 2) – (2x2 + 3x – 7)
= (4x2 + x + 2) + (–2x2 – 3x + 7)
2
4.
2
Add the opposite.
= 4x – 2x + x – 3x + 2 + 7
Group like terms.
= 2x2 – 2x + 9
Combine like terms.
Subtract the polynomials.
a) (5x2 + 3x – 7) – (2x2 – 5x + 3)
b) (–3y2 + 2y + 1) – (– 6y2 – 8y – 6)
Add the opposite.
Group like terms.
Combine like terms.
Get Ready ● MHR 363
Name: _____________________________________________________
Date: ______________
Math Link
Landscape Design
Here is a landscape design for a property.
1.
4.5 m
The radius of the circular herb garden is
4.5 m. What is the area of the herb garden?
Round your answer to the nearest tenth.
A = πr 2
Herb
Garden
8.5 m
Driveway
← Formula
2m
3m
Patio
House
2.3 m
27 m
Pool 4 m
← Substitute
4m
9m
← Evaluate
The area is
2.
.
36 m
a) What is the area of the whole property?
b) The house has a square base. What is the
area of the base of the house?
← Formula →
← Substitute →
← Evaluate →
__________________________________
__________________________________
__________________________________
__________________________________
c) What fraction of the property does the house take up? Write your answer in lowest terms.
area of the base of house
=
area of property
Sentence: ___________________________________________________________________
364
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
3.
Date: ______________
The pool is in the shape of a square with a trapezoid attached to it. The water in the pool is
1.7 m deep. What is the volume of water in the pool? Round your answer to the nearest tenth.
V = area of base × depth of water
a=2
Area of base of the pool = area of trapezoid + area of square
h = 2.3
= [h(a + b) ÷ 2] + (s2)
=
Substitute.
=
Evaluate.
b=4
s=4
V = area of base × depth of water
=
Substitute.
=
Evaluate.
Sentence: ________________________________________________________________________
4.
What is the area of the driveway? Round your answer to the nearest tenth.
Sentence: ________________________________________________________________________
5.
The area of the patio is 18 m2. What is the total area of the grass? Show your work.
Area of grass = A of property – A of pool – A of herb garden – A of house – A of patio – A of driveway
Sentence: ________________________________________________________________________
Math Link ● MHR 365
Name: _____________________________________________________
Date: ______________
7.1 Warm Up
1.
Write each expression as a single power. Use exponent laws.
a)
23 × 24
b) 52 × 5
= 23 + 4
=2
c) 44 × 44
2.
d) 102 × 103
Write each expression as a single power. Use exponent laws.
a)
25 ÷ 23
b) 44 ÷ 44
5−3
=2
=2
c)
3.
Divide.
366
5
d) 7
7
+÷+=+
+÷-=-
-÷- =+
-÷+ =-
a) 18 =
3
b) −20 =
5
21
=
−7
d) −36 =
−9
c)
4.
56
53
Multiply.
+×+=+
+×-=-
-×-=+
-×+=-
a) (–3)(10) =
b) (6)(7) =
c) (–5)(– 8) =
d) (9)(– 4) =
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Exponent laws:
am × an = am + n am ÷ an = am – n
(a m)n = a mn
(a × b)m = a m × bm
n
a
an
= n
a0 = 1, a ≠ 0
b
b
()
Name: _____________________________________________________
Date: ______________
7.1 Multiplying and Dividing Monomials
Link the Ideas
Literacy Link
Working Example 1: Multiply Monomials
monomial
• has 1 term
• examples: 5, 2x, 3s2,
n4
– 8cd,
3
Find each product.
a) (5x)(2x)
Solution
Method 1: Use a Model
Use x-tiles and x2-tiles to model (5x)(2x).
• Draw 5 positive x-tiles on 1 side.
• Draw 2 positive x-tiles on the other side.
You can use algebra
tiles to model
algebraic expressions.
positive x-tile
positive y-tile
positive xy-tile
2
Each square has an area of (x)(x) = x .
2
• Complete the rectangle with positive x -tiles.
positive x2-tile
The same tiles in
white represent
negative quantities.
positive x2-tiles.
There are
So, (5x)(2x) =
x2
Method 2: Use Algebra
Use the exponent law: x1 × x1 = x1 + 1
(5x)(2x)
= (5)(2)(x1)(x1)
=
x2
Multiply the numerical coefficients and variables.
Simplify.
7.1 Multiplying and Dividing Monomials ● MHR 367
Name: _____________________________________________________
b) (3x)(2y)
Solution
Method 1: Use a Model
Use x-tiles, y-tiles, and xy-tiles to model (3x)(2y).
Each grey rectangle inside the frame has an area of (x)(y) = xy.
xy-tiles.
There are
So, (3x)(2y) =
xy.
Method 2: Use Algebra
(3x)(2y)
= (3)(2)(x)(y)
Multiply the numerical coefficients and variables.
=
Simplify.
xy
Use a model to find each product. Then, use algebra to solve.
a) (4x)(2y)
Use a Model:
Use Algebra:
b) (− x)(7x)
Use a Model:
368
Use Algebra:
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Date: ______________
Name: _____________________________________________________
Date: ______________
Working Example 2: Apply Monomial Multiplication
Write an expression for the area of the rectangle.
4.3x
2x
Solution
Area = length × width
A = (4.3x)(2x)
A = (4.3)(2)(x)(x)
(4.3)(2) = 4.3 × 2
A=(
)(x1)(x1)
A=
x2
Group like terms.
An expression for the area of the rectangle is
.
Simplify the expressions by finding the product.
a) (−5x)(3.2)
(11a)(2b)
b)
=(
)(
=
ab
c) (2y)(4.4y)
d) (12h)(3.2h)
e) (6.2x)(x)
f ) (4.1y)(3z)
)ab
7.1 Multiplying and Dividing Monomials ● MHR 369
Name: _____________________________________________________
Date: ______________
Working Example 3: Divide Monomials
Find each quotient.
The quotient is the answer
when you divide.
−10 x 2
2x
a)
Solution
Method 1: Use a Model
numerator
denominator
Use algebra tiles to divide.
• Draw 2 positive x-tiles on the left side.
negative x2-tiles to represent the
• Draw
numerator.
Arrange the tiles into a rectangle so one side is 2 x-tiles
long.
• The other side is the unknown side length of the rectangle.
It is made up of 5 negative x-tiles.
−10 x 2
=
2x
So,
x
Method 2: Use Algebra
−10 x 2
2x
−5
−10 x 2
=
2 x1
Divide the numerator and the denominator by 2.
1
=
−
= –5x
x2
x
2−1
Use the exponent law to divide the variables.
= –5x
= –5x
370
MHR ● Chapter 7: Multiplying and Dividing Polynomials
?
Name: _____________________________________________________
Date: ______________
8 xy
4x
b)
Solution
Method 1: Use a Model
• Draw 4 positive x-tiles on the top.
?
• Draw
xy-tiles to represent the numerator.
Arrange the 8 xy-tiles into a rectangle so that 1 side is
4 x-tiles long.
• The other side is the unknown side length of the rectangle.
It is made up of 2 y-tiles. So,
8 xy
=
4x
y
Method 2: Use Algebra
8 xy
4x
2
8 xy
=
4x
Divide the numerator and the denominator by 4.
1
xy
=
x
1
xy
=
x
Divide the numerator and the denominator by x.
You can also use the
exponent law to divide
the variables.
1
=
y
Find each quotient by simplifying the expression.
a)
12 xy
3y
b)
−14 x 2
−2 x
7.1 Multiplying and Dividing Monomials ● MHR 371
Name: _____________________________________________________
Date: ______________
Working Example 4: Apply Monomial Division
The area of a triangle is given by the expression 18x2.
The base of the triangle is 4x.
What is the height of the triangle in terms of x?
Solution
Area = 18x 2
base × height
2
Area = base × height ÷ 2
4x
Rewrite the equation:
Height = 2 ×
=
area
base
2 18 x 2
×
4x
1
(2)(18x 2 )
=
4x
Substitute.
Multiply.
x2
=
4x
9
=
x
2
36 x
4x
Divide the numerator and denominator by 4.
Use the exponent law to divide the variables.
1 1
=
9
36 x 2−1
4
x
1
The height of the triangle is
.
Calculate each quotient by simplifying the expression.
a)
18 x 2
3x
14y ÷ (–2)
b)
=
14 y
=
c)
372
4.2 y
2
d)
−18.6mn
−3n
MHR ● Chapter 7: Multiplying and Dividing Polynomials
+÷+=+
+÷-=-
-÷- =+
-÷+ =-
Name: _____________________________________________________
Date: ______________
Check Your Understanding
Communicate the Ideas
1.
Show 2 ways you can find the product of (3x) and (5x).
2.
Laurie made a mistake in the question below.
a) Circle Laurie’s mistake(s).
(3x)(5x)
b) Show how to find the correct solution.
16n 2
2n
= (16 – 2)(n2 – n)
= 14n
Practise
3.
Write the multiplication statement for each set of tiles.
a)
b)
positive x-tile
positive y-tile
(2x)(
c)
)=
x2
(–
)(
)=
positive xy-tile
d)
positive x2-tile
The same tiles in
white represent
negative quantities.
7.1 Multiplying and Dividing Monomials ● MHR 373
Name: _____________________________________________________
4.
Model and complete each multiplication statement.
a) (2x)(4x)
b) (–3x)(2x)
‰ Draw 2 positive x-tiles on 1 side.
‰ Draw 4 positive x-tiles on the other side.
2
‰ Complete the rectangle with positive x -tiles.
(2x)(4x) =
5.
x2
Multiply.
(2y)(5y)
a)
= (2)(
b) (3a)(– 6b)
)( y)( y)
=
c) (– q)(–5q)
6.
d) (1.5m)(–3n)
Write an expression for the area of the rectangle.
3.9x
5x
374
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Date: ______________
Name: _____________________________________________________
7.
Date: ______________
Write a division statement for each set of tiles.
a)
b)
?
=
?
6 x2
=
x
c)
d)
?
?
=
=
8.
Model and complete each division statement.
a)
8x2
2x
b)
−9 x 2
−3 x
‰ Draw 2 positive x-tiles on the left side.
2
‰ Draw 8 positive x -tiles arranged in 2 rows.
‰ Find the unknown side.
−9 x 2
=
−3x
8x2
=
2x
x
7.1 Multiplying and Dividing Monomials ● MHR 375
Name: _____________________________________________________
9.
Find the quotient.
The quotient is the answer when you divide.
25st
5s
a)
Date: ______________
Divide the numerator and denominator by the coefficient, 5.
st
=
Divide the numerator and denominator by the variable, s.
s
=
b)
7x 2
x
c)
−8m
−2m
Apply
10. Write an expression for the area of each figure.
a)
Area of a triangle = b × h ÷ 2
b)
3x
4p
11x
10p
The expression is
__________________________________
.
11. Find the missing dimension in each figure.
Width =
a)
A=
15x 2
area
length
b)
1.1w
A = 3.3w 2
Base = 2 ×
area
height
?
5x
?
__________________________________
376
__________________________________
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
Date: ______________
12. The length of Jean’s rectangular patio is 20d.
b) The area of the patio is 72d 2.
Write an expression for its width.
a) Draw a rectangle and label the length.
Math Link
Landscape designers use different shapes for their designs,
such as rectangles or circles.
The shapes can show lawns, flower gardens, patios, or pools.
a) Choose what the rectangle and circle will show in your design.
Label each shape.
Using variables, label the length and width of the rectangle and the radius of the circle.
b) Write an area formula for each shape.
rectangle:
c)
circle:
What type of material will you use to fill each shape?
rectangular prism:
cylinder:
How deep or how high.
d) What is the depth of each shape?
rectangular prism:
e)
cylinder:
Write a volume formula for each shape.
rectangular prism:
cylinder:
7.1 Math Link ● MHR 377
Name: _____________________________________________________
7.2 Warm Up
1.
Write the multiplication statement for each model.
a)
b)
(
2.
x)(
)=
x2
Draw a model for each multiplication expression. Then, write the product.
a) (5x)(2x)
b) (–3x)(–2x)
‰ Draw 2 positive x-tiles on the left side.
‰ Draw 5 positive x-tiles on the top.
‰ Complete the rectangle with positive
x2-tiles.
So, (5x)(2x) =
3.
378
Multiply.
a) (3n)(5n)
b) (– 4y)(6y)
c) (–3.2w)(2w)
d) (t)(–7t)
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Date: ______________
Name: _____________________________________________________
Date: ______________
7.2 Multiplying Polynomials by Monomials
Link the Ideas
Working Example 1: Multiply a Polynomial by a Monomial Using an Area Model
Find the product of (3x)(2x + 4).
Literacy Link
polynomial
● made up of terms
connected by addition or
subtraction
● examples: x + 5, 2d – 2.4
Solution
2x + 4
Draw a rectangle. Label the sides 3x and 2x + 4.
3x
Divide the large rectangle into 2 smaller rectangles.
2x
2x + 4
4
Calculate the area of each small rectangle.
A1 = (3x)(2x)
3x
A2 = (3x)(4)
x2
A1 =
A2 =
A1
A2
x
total area
Total area = A1 + A2
x2 +
=
A1= 6x 2
x
A2= 12x
Calculate each product.
a) (2x)(x + 3)
x
2x
b) (2 + c)(c)
x+3
A1
3
A2
A1 = (
A1
x)(x)
A2 = (
x2
A1 =
A2
x)(3)
A2 =
x
Total area = A1 + A2
=
x2 +
x
7.2 Multiplying Polynomials by Monomials ● MHR
379
Name: _____________________________________________________
Date: ______________
Working Example 2: Multiply a Polynomial by a Monomial Using Algebra Tiles
Find the product of (2x)(3x – 5).
Solution
-1 + x - x
+ x2
Use positive x-tiles and negative 1-tiles to model 2x and 3x – 5.
Use positive x2-tiles and negative x-tiles to complete the rectangle.
There are
positive x2-tiles and
So, (2x)(3x – 5) =
x2 –
negative x-tiles.
x
Find each product.
a) (3x + 2)(3x)
b) (4x)(2x – 1)
‰ Draw 3 positive x-tiles and 2 positive
1-tiles along the top.
‰ Draw 3 positive x-tiles along the side.
2
‰ Use x -tiles and positive x-tiles to
complete the rectangle.
(4x)(2x – 1) =
So, (3x + 2)(3x) =
380
x2 +
x
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
Date: ______________
Working Example 3: Multiply a Polynomial by a Monomial Algebraically
The expressions 4x and 5x – 3 show the dimensions of a rectangular gym.
Find the polynomial expression for the area of the floor.
Write the expression in simplified form.
4x
Solution
5x - 3
Find the area of the rectangle.
A = length × width
distributive property
● used to simplify algebraic expressions
● multiply each term in the polynomial by the monomial
●
a(b + c) = ab + ac
Use the distributive property. Multiply each term in the binomial 5x – 3 by 4x.
binomial
● a polynomial with
2 terms
2
● examples: 6y + 3,
2x – 5
A = (4x)(5x – 3)
A = (4x)(5x) – (4x)(3)
A = (4)(5)(x)(x) – (4)(3)(x)
x2 –
A=
Literacy Link
x
A simplified expression for the gym floor is 20x2 –
x.
Calculate each product.
(–3x)(2x + 5)
a)
= (–3x)(
b) (5y)(11 – x)
) + (–3x)(
)
=
c) (2a)(–3a + 8)
d) (6k)(–5 – 3k)
7.2 Multiplying Polynomials by Monomials ● MHR 381
Name: _____________________________________________________
Date: ______________
Check Your Understanding
Communicate the Ideas
1.
Show 2 different ways to simplify the expression (3x)(2x + 4).
2.
Mahmoud expanded the expression (5x)(2x + 1).
(5x)(2x +1 ) = 10x2 + 1
a) Is his work correct? Circle YES or NO.
b) Give 1 reason for your answer.
_____________________________________________________________________________
c) Write the correct answer.
Practise
3.
Write the multiplication expression for each model.
a)
2x
b)
4
4k
3k
3x
3.6
3x(
382
+
)
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
4.
Expand each expression using an area model.
Expand means multiply the first
term by each term in the brackets.
a) (2x)(4x + 2)
2x
Date: ______________
b) (a)(3a + 6)
4x
2
A1
A2
A1 = (2x)(
A1
)
A2 = (2x)(
=
A2
)
=
Total area = A1 + A2
=
+
=
5.
Write the multiplication statement for the models.
a)
b)
(
=
c)
x)(
x+
x2 +
x
)
d)
7.2 Multiplying Polynomials by Monomials ● MHR 383
Name: _____________________________________________________
6.
Use algebra tiles to expand each expression.
a) (3x)(x – 2)
b) (2x)(–2x + 1)
‰ Draw 1 positive x-tile and
2 negative 1-tiles along the top.
‰ Draw 3 positive x-tiles
along the side.
2
‰ Use x -tiles and negative x-tiles
to complete the rectangle.
7.
Expand using the distributive property.
(2x)(3x – 1)
a)
= (2x)(
b) (3p)(2p + 5)
) – (2x)(
)
=
c) (4j)(2j – 3)
384
d) (0.1r)(30r + 10)
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Date: ______________
Name: _____________________________________________________
Date: ______________
Apply
8.
A rectangular blanket has a width of 3x and a length of 4x – 3.
a) Write an expanded expression for the area of the blanket.
3x
Area = length × width
4x – 3
b) Write a simplified expression for the perimeter of the blanket.
Perimeter = (2 × length) + (2 × width)
9.
A rectangular field is (4x + 2) metres long.
The width of the field is 2 m shorter than the length.
a) Write the expression for the width of the field:
b) Label the rectangle with the expressions for the length and width.
c) What is an expression for the area of the field?
Sentence: _____________________________________________________________________
7.2 Multiplying Polynomials by Monomials ● MHR 385
Name: _____________________________________________________
Date: ______________
Math Link
Draw a landscape design. Include 1 of the following in the shape of a rectangle:
• swimming pool
• concrete patio
• hockey rink
The rectangle’s length is 2 m longer than twice the width.
a) Let w represent the width.
Write a polynomial to represent the length of the rectangle:
b) How much higher or lower than ground level will your design be?
This is the depth.
m
A swimming pool could have a depth of 1.5 m.
c)
Write a formula for calculating the volume of your design.
V = length × width × height
Use the depth from part b).
d) Complete the table. Find the volume of material needed for widths of 2 m, 3 m, 4 m, and 5 m.
An example using a width of 1 m and a depth of 0.8 m has been done for you.
Use the depth from part b).
Width (w)
Example:
1m
Length (2w + 2)
Depth (m)
V=l×w×h
2(1) + 2
=2+2
=4
0.8
4 × 1 × 0.8
= 3.2 m3
2m
3m
4m
5m
5.
Which width works best for your design?
Give 1 reason for your answer.
________________________________________________________________________________
386
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
Date: ______________
7.3 Warm Up
1.
Write a division statement for each set of algebra tiles.
a)
b)
?
?
x2 ÷
x=
x
c)
2.
Model and complete each division statement.
a)
3.
d)
6 x2
=
2x
x
b)
−12 x 2
=
4x
Find the quotient.
35 p 2
a)
−7 p
−25m 2
b)
−5
7.3 Warm Up ● MHR 387
Name: _____________________________________________________
Date: ______________
7.3 Dividing Polynomials by Monomials
Link the Ideas
Working Example 1: Divide a Polynomial by a Monomial Using a Model
Find the quotient of
6x2 − 8x
.
2x
Solution
Use algebra tiles.
● The left side represents the denominator or divisor, 2x.
● Draw 2 positive x-tiles.
●
?
Model the numerator, 6x2 – 8x.
?
2
Use
positive x -tiles and
negative x-tiles.
Make a rectangular shape to match the left side.
●
Count the number of positive x-tiles and negative 1-tiles
you need to complete the top.
There are
positive x-tiles and
negative 1-tiles, or 3x – 4.
So,
6x2 − 8x
= 3x –
2x
Check:
Multiply the quotient, 3x – 4, by the divisor, 2x.
(2x)(3x – 4)
= (2x)(3x) – (2x)(
)
You can also find the
quotient using algebra.
6x 2 – 8x
2x
2
6x
8x
=
–
2x
2x
3
=
2
x –
x
The answer is correct. It matches your numerator, or dividend.
388
MHR ● Chapter 7: Multiplying and Dividing Polynomials
=
6 x 2--1
1
2
= 3x – 4
4
–
8 x 1--1
12
Name: _____________________________________________________
Date: ______________
Use a model to find each quotient.
a)
3x 2 + 6 x
3x
‰ Model 3x on the left side.
‰ Model the numerator.
Use
positive x2-tiles and
positive x-tiles.
‰ Make a rectangular shape to match the left side.
‰ Count the number of positive x-tiles and negative 1-tiles you need to complete the top.
Draw them.
So,
b)
3x 2 + 6 x
=
3x
x+
8x2 − 2 x
2x
So
8x2 − 2 x
=
2x
x−
7.3 Dividing Polynomials by Monomials ● MHR 389
Name: _____________________________________________________
Date: ______________
Working Example 2: Dividing a Polynomial by a Monomial Algebraically
a) Find the ratio of the surface area to the radius of the cylinder.
Write the ratio in simplified form.
r
Solution
surface area 2πr 2 + 2πrh
=
radius
r
=
2πr 2 2πrh
+
r
r
Divide each term by r.
= 2πr2−1 + 2πr1−1h
Use the exponent law to divide the variable, r.
= 2πr +
b) Let the height, h, of the cylinder be the same as the radius, r.
Find the ratio of the surface area to the radius.
Write the ratio in simplified form.
Solution
surface area
= 2πr + 2πh
radius
= 2πr + 2π(r)
Use the answer from part a).
Substitute h = r.
= 2πr + 2πr
=
πr
Find each quotient.
a)
15 x 2 − 12 x
3x
=
b)
15x 2
–
x
=
390
12x
x
x–
MHR ● Chapter 7: Multiplying and Dividing Polynomials
−2t 2 + 4t
2t
h
Name: _____________________________________________________
Date: ______________
Check Your Understanding
Communicate the Ideas
1.
List the steps to model
4 x2 + 6 x
.
2x
‰ _____________________________________________________________________________
‰ _____________________________________________________________________________
‰ _____________________________________________________________________________
_____________________________________________________________________________
2.
Anita simplified an expression.
a) Circle Anita’s mistake.
b) How would you correct her mistake?
9k 2 - 3k
3
9k 2 3k
=
3
3
= 3k - 1
__________________________________
__________________________________
__________________________________
Practise
3.
Write the division statement for the algebra tiles.
Then, find the quotient.
Quotient =
a)
tiles on left side
b)
?
x2 +
tiles under the frame
?
x
=
x+2
x
7.3 Dividing Polynomials by Monomials ● MHR 391
Name: _____________________________________________________
4.
Date: ______________
Write the division statement for the algebra tiles. Then, find the quotient.
a)
b)
?
?
__________________________________
__________________________________
5.
Use a model to divide each expression.
a)
4 x 2 + 12 x
2x
b)
− 6 x 2 − 3x
3x
‰ Model 2x on the left side.
‰ Model the numerator.
Use
positive x2-tiles and
positive x-tiles.
‰ Make a rectangular shape to match the left side.
‰ Count the number of positive x-tiles and
1-tiles to complete the top. Draw them.
So,
392
4 x 2 + 12 x
=
2x
x+
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
6.
Date: ______________
Divide.
2 y2 + 4 y
2y
a)
=
b)
−18 y 2 − 6 y
−6 y
d)
2.7c 2 + 3c
3c
2 y2
4y
+
2y
2y
=y+
c)
2 x 2 + 8 xy
x
Apply
7.
A rectangle has an area of 9x2 – 3x square units.
The length of the rectangle is 3x units.
What is the width?
width =
area
length
?
9x 2-3x
3x
Sentence: _______________________________________________________________________
8.
The grade 9 students want to decorate the gym wall for a dance.
The expression 45x2 + 20x represents the area of the wall.
The expression 5x represents the area covered by 1 sheet of poster paper.
What expression represents the number of sheets of poster paper that will cover the wall?
Draw a diagram to help you.
Sentence: _______________________________________________________________________
7.3 Dividing Polynomials by Monomials ● MHR 393
Name: _____________________________________________________
9.
Date: ______________
A rectangular fish tank has the dimensions shown, in metres.
The volume of the tank is represented by 7.5w2 – 3w.
a) What expression represents the area of the base?
Area of base =
volume
height
l
0.6 m
w
Sentence: ____________________________________________________________________
b) The width of the base is represented by w.
What expression represents the length of the tank?
length =
area of the base
width
Use your answer from part a).
Sentence: ____________________________________________________________________
c) What is the length of the tank if the width is 0.6 m?
Use your answer from part b).
Substitute w = 0.6.
Sentence: ____________________________________________________________________
d) What is the volume of the tank if the width is 0.6 m?
Sentence: ____________________________________________________________________
394
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
Date: ______________
Math Link
You are designing a park. It includes a large parking lot that will be covered with gravel.
a) Design 2 different rectangular parking lots.
The lengths must be between 50 m and 100 m and the widths between 40 m and 65 m.
Draw your parking lots and label the dimensions.
Parking Lot 1:
Parking Lot 2:
b) Calculate the area of each parking lot.
Parking Lot 1:
c)
Parking Lot 2:
A truck with these dimensions delivers the gravel.
Write an expression for the volume of the truck.
xm
(x + 4) m
V=l×w×h
V=(
1m
)(
)(1)
V=
d) The depth of gravel in each parking lot will be
0.05 m. Write an expression for the area that
1 load will cover.
Area of 1 load =
e)
The width, x, of the truck delivering the
gravel is 2 m. How many square metres
of gravel will 1 load cover?
volume of truck
depth of gravel
Substitute x = 2 into your
answer from part d).
One load of gravel will cover
f)
How many truckloads of gravel will you need for each parking lot?
Parking Lot 1:
Parking Lot 2:
m2.
area of parking lot
area of 1 load of gravel
7.3 Math Link ● MHR 395
Name: _____________________________________________________
Date: ______________
Graphic Organizer
Simplify each expression. Show different methods.
Use a model.
(3x)(2x)
Use algebra.
(–4x)(5x)
ials
Divid
onom
ing M
iply
Mult
ing M
onom
ials
Use a model.
– 6x2
2x
Use algebra.
16xy
4x
Multiplying and
Dividing
Polynomials
nomi
y Mo
ials b
g Pol
ynom
396
Mult
MHR ● Chapter 7: Multiplying and Dividing Polynomials
ials
Use algebra.
(2x)(3x + 1)
om
Mon
iplyin
ls by
omia
olyn
ing P
Use algebra tiles.
(2x)(3x + 1)
Use algebra tiles.
4x2 – 6x
2x
Divid
als
Draw a rectangle.
(2x)(3x + 1)
Use algebra.
4x2 – 6x
2x
Name: _____________________________________________________
Date: ______________
Chapter 7 Review
Key Words
For #1 to #6, write the number of the polynomial in Column A beside the equivalent polynomial
in Column B.
Column A
Column B
1.
8 xy
2x
4xy – 2x
2.
12 x 2 − 6 x
3x
4x2 – 2x
3.
(–2x)(–2x + 1)
4y
4.
12 xy − 6 x
3
2x2 – 2x
5.
8 xy
2
4xy
6.
12 x 2 − 12 x
6
4x – 2
7.1 Multiplying and Dividing Monomials, pages 367–377
7.
Draw a model to complete the multiplication statement.
a) (2x)(4x)
b) (–3x)(3x)
‰ Draw 2 positive x-tiles on the top.
‰ Draw 4 positive x-tiles on the side.
‰ Complete the rectangle with x2-tiles.
(–3x)(3x) =
(2x)(4x) =
Chapter 7 Review ● MHR 397
Name: _____________________________________________________
8.
Find each product.
a) (– 8x)(11x)
9.
b)
(1.1x)(5x)
Draw a model to complete the division statement.
6x2
a)
2x
b)
−8 x 2
4x
‰ Draw 2 positive x-tiles on the left
side.
‰ Arrange 6 positive x2-tiles in 2 rows.
‰ Find the unknown side.
−8 x 2
=
4x
6x2
=
2x
10. Find each quotient.
2 xy
a)
2x
398
b)
− 4.2r 2
−2 r
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Date: ______________
Name: _____________________________________________________
Date: ______________
7.2 Multiplying Polynomials by Monomials, pages 379–386
11. Write the polynomial multiplication statement for the area model.
2y
3y
5
A1
A2
A1 = (
Area = length × width
)(
)
A2 =
=
Total area = A1 + A2
=(
)(
+
)
=
12. Write the polynomial statement for the algebra tiles.
a)
(3x)(
x+
x2 +
)=
b)
13. Use the distributive property to simplify.
(20x)(2x – 1)
a)
= (20x)(
=
b) (−3x)(1.2x + 6)
) – (20x)(
)
–
Chapter 7 Review ● MHR 399
Name: _____________________________________________________
Date: ______________
7.3 Dividing Polynomials by Monomials, pages 388–395
The quotient is the
answer when you divide.
14. Write the division statement for the algebra tiles. Give the quotient.
a)
b)
?
(
x2 +
)÷x=
?
+
15. Divide.
a)
12n 2 − 2n
2n
2
12n
2n
–
=
2n
2n
=
15 x − 3x 2
3x
–
16. A triangle has an area represented by 3x2 + 6x.
The base of the triangle is 3x. Find the height.
Height = 2 ×
b)
Draw a diagram
to help you.
area
base
Sentence: ________________________________________________________________________
400
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
Date: ______________
Key Word Builder
Use the clues to unscramble each key word.
Clue
Scrambled Word
a) An expression formed by adding or subtracting
terms, such as 3s2 – 6x + 5.
lyoploinam
b) 3x(2x – 8) = 6x2 – 24 is an example of using the
edtbuirtsiiv yppoetrr
Key Word
.
(2 words)
c)
An expression with 2 terms, such as 6y2 + 4.
d) An expression with 1 term, such as 5x.
oiinmabl
alnoommi
Solve the expressions.
Match the letter of each answer to the blanks below to find the missing word.
L 3x(3x – 3)
I
(3x)(2y)
O
16 x 2 − 4 x
4x
A (x)(2x)
N
18 x 2
x2
M
15 xy
3 xy
Mae’s answer was a ___
5
___
4x – 1
___
18
___
4x – 1
___
5
___
6xy
___
2x2
___ when she divided.
9x2 – 9x
Key Word Builder ● MHR 401
Name: _____________________________________________________
Date: ______________
Chapter 7 Practice Test
For #1 to #6, circle the best answer.
1.
2.
3.
4.
5.
6.
402
Which monomial multiplication statement does the model show?
A (3x)(–2x) = – 6x2
B (2x)(–3x) = – 6x2
C (2x)(3x) = 6x2
D (–2x)(–3x) = 6x2
Multiply
What is the product of 3y and 2.7y?
A 0.9y
B 8.1y
C 0.9y2
D 8.1y2
Which monomial statement does the model show?
A
−6 x 2
= –2x
−3 x
B
−6 x 2
= 2x
−3 x
C
6x2
= –2x
−3x
D
6x2
= 2x
−3x
−27 q 2
What is the quotient of
?
9q
A 3q2
B 3q
C –3q
D –3q2
What is the product of (3x)(–3x – 6)?
A –9x2 + 18x
B –9x2 – 18x
C 9x2 + 18x
D 9x2 – 18x
Calculate
15 y 2 − 10 y
.
−5 y
A –3y – 2
B –3y + 2
C 3y – 2
D 3y + 2
MHR ● Chapter 7: Multiplying and Dividing Polynomials
?
Name: _____________________________________________________
Date: ______________
Short Answer
For #7 to #10, show your work.
7.
Calculate (1.3x)(4y).
8.
Find the product of (12h)(–3h + 2).
9.
Simplify
−3 x 2 + 3 x
.
−3 x
Chapter 7 Practice Test ● MHR 403
Name: _____________________________________________________
10. Paula is building a rectangular patio.
It will have a square flower bed in the middle.
Date: ______________
A = 3.5x 2
4x
3.1x
a) The length of the patio is 4x.
The width is 3.1x.
Write an expression for the area.
b) The area of the flower bed is 3.5x2.
The rest of the patio will be paving stones.
What area of the patio needs paving stones?
__________________________________
__________________________________
11. A sports field is 15 m longer than twice the width.
a) Let w represent the width of the field.
Write an expression for the length.
404
b) Write an expression for the area of the
field. Use the distributive property to
simplify the expression.
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
Date: ______________
Math Link: Wrap It Up!
You have been hired to create a landscape design for a park.
It must include a
● rectangular sand area for beach volleyball
A beach volleyball court
● cylindrical wading pool
is 9 m × 18 m.
a) What are the dimensions of each feature in metres?
Volleyball court:
Wading pool:
length =
m
width =
m
radius =
b) How deep is the sand for the volleyball court?
How deep is the water in the wading pool?
c)
m
m
m
Draw the park design on a separate sheet of paper. Label all the dimensions.
d) Write a polynomial expression for each area.
Volleyball court:
width = x
length = 2x – 5
A = length × width
Wading pool:
r = 3y
A=π×r×r
A = π × (3y)(
)
A=π×(
)
A = 3.14(
)
A=
e)
Write a polynomial expression for each volume.
Use your answer from part d) for the area of the base.
Use your answer from part b) for the depth.
Volleyball court:
V = area of base × depth
Wading pool:
V = area of base × depth
Math Link: Wrap It Up! ● MHR 405
Name: _____________________________________________________
Date: ______________
Challenge
Polynomial Puzzle
1.
Try this 9-piece puzzle.
-3x + 2
(4x)(-5y)
9x__z___
____
(-3xz)
3x - 4
24x2
2
15
__x__
__10
__x_
-5x
5x(3 - 2y)
Materials
• Polynomial Puzzle BLM
per student
• Blank Puzzle Pieces BLM
per student
• scissors
-5y (2x + 3y)
2x
____4.__8x__2_
1.2x
y2
15x - 10xy
(-3x)(-8x)
(3yz)(4)
6x2 - 10x
-4x
-3
3xy + 9y
________
3y
-10xy - 15
2xy
_________
x+3
y
12yz
-20xy
2 - 12x
9x
________
3x
2x(3x - 5)
a) Cut out the puzzle pieces from Polynomial Puzzle BLM.
b) Solve the puzzle.
● Polynomial expressions involving multiplication or division are black.
● Matching equivalent expressions are grey.
Match each multiplication or division expression to its equivalent expression.
The diagram shows the pattern of the solution for a 9-piece puzzle.
2.
Design your own 9-piece puzzle.
a) On a separate sheet of paper, write 9 polynomial expressions using multiplication and division.
● Include the types of polynomial expressions from Chapter 7.
Include a variety of constants, monomials, and polynomials.
● Solve the polynomial expressions.
b) Write these matching expressions on the Blank Puzzle Pieces BLM.
Place them the same way as in the small diagram above.
Each expression and its equivalent must be across from each other.
c) Cut out your 9 puzzles pieces. Mix them up.
d) Exchange your puzzle with a classmate’s. Solve each other’s puzzles.
406
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
Date: ______________
Chapters 5–7 Review
Chapter 5 Introduction to Polynomials
1.
Draw algebra tiles to collect like terms in each expression.
Write the simplified expression.
a) –1 + x – 1 – x + 1
positive 1-tile
negative 1-tile
A 1-tile has dimensions 1 unit by 1 unit.
positive x-tile
negative x-tile
An x-tile is 1 unit by x units.
negative x2-tile
positive x2-tile
An x2-tile is x units by x units.
b) g2 – g + 5 + 2g – 4g2
2.
Simplify the expressions. Write each expression in simplest form.
b) (w2 – 4w + 7) + (3w2 + 5w – 3)
(2m – 3) + (5m + 1)
a)
= 2m – 3 + 5m + 1
Remove brackets.
= 2m +
Group like terms.
=
m–3+
m–
Combine like terms.
d) (3d 2 – 7) – (d 2 – 2)
c) (–7z + 3) – (– 4z + 5)
Add the opposite.
Remove brackets.
Group like terms.
Combine like terms.
Chapters 5 –7 Review ● MHR 407
Name: _____________________________________________________
3.
Date: ______________
A park is divided equally into 3 square sections. Each section has a side measurement of 2n + 4.
The solid lines show where a new fence will be built. The openings in the fence all measure n.
2n + 4
n
a) Write the simplified expression for the length of the fence on 1 side of the square.
b) How many sides will be fenced?
c) Write an expression for the total length of the fence:
d) Write a simplified expression for the total length of the fence.
4.
The Better Buys antique shop sells comic books for $10,
hardcover books for $8, and paperback novels for $3.
$8
$3
a) Write an algebraic expression for the income from
each type of book. Choose a different variable for each.
Type of Income
Variable
Expression
Comic book
Hardcover book
Paperback novel
b) Write an algebraic expression for the total income from the sale of comic books,
hardcover books, and paperback novels.
408
MHR ● Chapter 7: Multiplying and Dividing Polynomials
$10
Name: _____________________________________________________
Date: ______________
Chapter 6 Linear Relations
5.
Figure 1
Figure 2
Figure 3
a) The pattern starts with
tiles.
Then, add
tiles each time
to make the next figure.
b) Complete the table of values.
Figure Number, n Number of Tiles, t
1
2
3
4
c) What equation relates the number of tiles
to the figure number?
6.
d) If the pattern continues, how many tiles
will be in Figure 8?
Monika is saving money for a ski trip. She starts with $112 in her bank account.
She deposits $25 in her account every week.
a) Complete the table of values.
b) Complete the equation to model this
situation:
Week, w
Amount in the
Bank, A ($)
0
$112
1
2
A=
w+
c) Monika needs $450 for her ski trip. How
many weeks will it take for her to save up
this money?
3
4
5
Chapters 5 –7 Review ● MHR 409
Name: _____________________________________________________
7.
Date: ______________
Car mechanics charge $35 to start a job. Then, they charge $60/h to fix the problem.
So, mechanics are paid a base fee of $35 plus $60/h. The graph shows this linear relation.
C
Repair Costs
500
450
400
Cost ($)
350
300
250
200
150
100
50
0
1
2
3
4
5
6
7
8 t
Time (h)
a) Approximately how much would a mechanic charge after working on a vehicle for 8 h?
Extend the line
on the graph.
b) Approximately how many hours would a mechanic work to charge $225?
Chapter 7 Multiplying and Dividing Polynomials
8.
Find the product.
(3x)(4x)
a)
b) (2.5y)(–2y)
= (3)(4)(x)(x)
=
9.
Divide.
−12h 2
2h
a)
Divide the top and bottom by 2.
−
=
b)
8.4 x 2
x
h2
Divide the top and bottom by h
or use the exponent laws.
h
=
10. Use the distributive property to simplify each expression.
(3x)(6x – 5)
a)
= (3x)(6x) – (3x)(
=
410
b) (1.5w)(2w + 1)
)
–
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
Date: ______________
11. a) A foosball table is 3 cm longer than twice its width.
The width of the table is represented by w.
Complete the expression for the length.
Length =
w+
b) What is the expression for the area of the foosball table?
Write your answer in expanded form.
Sentence: _____________________________________________________________________
12. Find each quotient.
12 g 2 + 8 g
4g
a)
=
b)
− 6 x 2 + 3xy
3x
12 g 2
8g
+
4g
4g
=
+
13. The area of a rectangle is represented by the expression 10x2 – 5x.
The length of the rectangle is 5x. Write an expression for the width.
Then, simplify the expression.
width =
area
length
A = 10x 2 – 5x
?
5x
Sentence: ________________________________________________________________________
Chapters 5 –7 Review ● MHR 411
Name: _____________________________________________________
Date: ______________
Task
Choosing a Television to Suit Your Room
1.
Materials
• measuring tape
• grid paper
To choose the best size of TV for a room, you need know the
distance from where you are sitting to the TV.
The table lists the best viewing distance for
This is the viewing distance.
a standard TV.
Screen Size
(cm)
Viewing Distance
of Standard TV (cm)
68.8
205.7
81.3
243.8
94.0
281.9
a) To find the best size of standard TV for a room, multiply the viewing distance by
1
.
3
A TV is 330 cm away from your seat. Calculate the best size of TV for the room.
Sentence: _____________________________________________________________________
b) On a separate sheet of paper, make a drawing of your classroom.
Include
• where you would put the TV
• where you will sit to have the best view
c) Measure the actual distance, in cm, from the TV to where you will sit to have the best view.
cm
d) What size of TV would be best for your classroom? Round your answer to the nearest
centimetre.
Sentence: _____________________________________________________________________
412
MHR ● Chapter 7: Multiplying and Dividing Polynomials
Name: _____________________________________________________
2.
Date: ______________
The diagram shows the viewing angle for different types of TVs.
Calculate the viewing area in square metres of a standard TV with a screen size of 94.0 cm.
r = viewing distance
=
cm
=
m
Area of circle = πr2
r
100 cm = 1 m
standard
plasma
LCD
r = viewing distance
Viewing angle:
standard: 120°
plasma: 160°
LCD: 170°
Fraction of circle covered by viewing angle =
=
=
viewing angle in degrees for standard TV
number of degrees in a circle
360
Write in lowest terms.
Viewing area of television = area of circle × fraction of circle covered
=
The viewing area of a 94.0 cm standard TV is
3.
.
Would this be the best type of TV and size for your classroom? Circle YES or NO.
Give 1 reason for your answer.
___________________________________________________________________
Task ● MHR 413
Answers
Get Ready, pages 362–363
7.1 Multiplying and Dividing Monomials, pages 367–377
1. Complete the table.
Working Example 1: Show You Know
Type of Polynomial
(Monomial, Binomial, or Trinomial)
Degree of
Polynomial
trinomial
2
a) x2 – 2x + 5
b) 11c + 14
binomial
1
c) 24d2
monomial
2
b) –7x2
a) 8xy
2. a) 3x2 – x b) 2g2 – 4n
3. a) 2x2 + x – 4
b) 2x2 + x + 1
Working Example 2: Show You Know
-
a) –16x b) 22ab c) 8.8y2 d) 38.4h2 e) 6.2x2 f) 12.3yz
Working Example 3: Show You Know
a) 4x b) 7x
Working Example 4: Show You Know
a) 6x b) –7y c) 2.1y d) 6.2m
Communicate the Ideas
1. (3x) × (5x) = 15x2
4. a) 3x2 + 8x – 10 b) 3y2 + 10y + 7
Math Link
1. 63.6 m2
2. a) The area is 972 m2. b) The area is 81 m2.
1
of the property.
c) The house takes up
12
3. The volume of water in the pool is 38.9 m3.
4. The area of the driveway is 25.5 m2.
5. The area of grass is 761 m2.
7.1 Warm Up, page 366
1. a) 27 b) 53 c) 48 d) 105
16n 2
= 8n
2n
2. a) 22 b) 40 c) 53 d) 74
2. a) (16 – 2)(n2 – n) b)
3. a) 6 b) – 4 c) –3 d) 4
Practise
4. a) –30 b) 42 c) 40 d) –36
3. a) (2x)(3x) = 6x2 b) (–2x)(3x) = – 6x2 c) (–2x)(–2x) = 4x2
d) (x)(2y) = 2xy
4. a) 8x2
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MHR ● Chapter 7: Multiplying and Dividing Polynomials
b) –6x2
5. a) 10y2 b) –18ab c) 5q2 d) –4.5mn
7.2 Multiplying Polynomials by Monomials, pages 379–386
6. (3.9x)(5x) = 19.5x2
2
7. a)
6x
3x
Working Example 1: Show You Know
− 6x
− 6x
8 xy
= 4x d)
= –3x c)
= 3x
2y
2x
−2 x
2
= 2x b)
2
a) 2x2 + 6x b) 2c + c2
Working Example 2: Show You Know
a) 9x2 + 6x;
8. a) 4x;
b) 8x2 – 4x;
b) 3x;
Working Example 3: Show You Know
a) –6x2 – 15x b) 55y – 5xy c) – 6a2 + 16a d) –30k – 18k2
Communicate the Ideas
1. Answers will vary. Examples:
2x + 4
9. a) 5t b) 7x c) 4
3x
2x
4
A1
A2
A1 = 6x2
Apply
A2 = 12x
10. a) A = 33x2 b) The expression is A = 20p2.
11. a) The missing dimension is 3x. b) The missing dimension is 6w.
12. a)
(3x)(2x + 4)
= (3x)(2x) + (3x)(4)
= 6x2 + 12x
20d
b) The width is 3.6d.
Math Link
Answers will vary. Examples:
a) rectangle: pool; circle: garden
b) rectangle: A = l × w; circle: A = πr
2. a) NO b) Mahmoud did not multiply +1 by 5x. c) 10x2 + 5x
2
Practise
c) rectangular prism: water; cylinder: soil
3. a) (3x)(2x + 4) b) (4k)(3k + 3.6)
d) rectangular prism: 5 m; cylinder: 0.5 m
4. a) 8x2 + 4x b) 3a2 + 6a
2
e) rectangular prism: V = l × w × 5; cylinder: (πr )(0.5)
7.2 Warm Up, page 378
5. a) (2x)(3x + 1) = 6x2 + 2x b) (–x)(–x – 3) = x2 + 3x
c) (3x)(–x + 2) = –3x2 + 6x d) (–x)(x + 4) = –x2 – 4x
1. a) (3x)(3x) = 9x2 b) (–2x)(–2x) = 4x2
6. a) 3x2 – 6x;
b) – 4x2 + 2x;
2
2. a) 10x ;
b) 6x2;
7. a) 6x2 – 2x b) 6p2 + 15p c) 8 j 2 – 12 j d) 3r 2 + r
Apply
8. a) 12x2 – 9x b) P = 14x – 6
9. a) 4x b)
c) The area of the field is 16x2 + 8x.
4x + 2
4x
3. a) 15n2 b) –24y2 c) – 6.4w2 d) –7t2
Answers ● MHR 415
5. a) 2x + 6;
Math Link
b) –2x – 1;
Answers will vary. Examples:
a) 2w + 2 b) 3 c) V = 6w2 + 6w
d)
Width (w)
Example:
Length (2w + 2)
Depth (m)
0.8
V=l×w×h
4 × 1 × 0.8
= 3.2 m3
1m
2(1) + 2
=2+2
=4
2m
6
3
36 m3
3m
8
3
72 m3
4m
10
3
120 m3
5m
12
3
180 m3
5. The width of 5 m works best for my design because it’s a pool.
7.3 Warm Up, page 387
1. a) 6x2 ÷ 2x = 3x b) 9x2 ÷ (–3x) = (–3x) c) – 6x2 ÷ (–2x) = 3x
d) 4x2 ÷ (–2x) = (–2x)
2. a) 3x;
b) –3x;
6. a) y + 2 b) 3y + 1 c) 2x + 8y d) 0.9c + 1
Apply
7. The width is 3x – 1.
8. 9x + 4 represents the number of sheets of poster paper needed.
9. a) 12.5w2 – 5w represents the area of the base. b) 12.5w – 5 represents
the length. c) The length is 2.5 m. d) The volume is 0.9 m3.
Math Link
Answers will vary. Examples:
a) Parking Lot 1:
Parking Lot 2:
75 m
90 m
40 m
50 m
3. a) –5p b) 5m2
b) Parking Lot 1: 3750 m2; Parking Lot 2: 3600 m2 c) V = x2 + 4x
7.3 Dividing Polynomials by Monomials, pages 388–395
d)
Working Example 1: Show You Know
a) x + 2;
b) 4x – 1;
x2 + 4x
e) 240 m2 f) Parking Lot 1: 16 loads;
0.05
Parking Lot 2: 15 loads
Graphic Organizer, page 396
Multiplying Polynomials:
Use a model: (3x)(2x) = 6x2
Working Example 2: Show You Know
a) 5x − 4 b) –t + 2
Communicate the Ideas
1. Draw 2 positive x-tiles on the left side.
Draw 4 positive x2-tiles and 6 positive x-tiles inside the frame.
Draw 2 positive x-tiles and 3 positive 1-tiles along the top.
Use algebra: (–4x)(5x) = –20x2
Dividing Monomials:
Use a model:
− 6 x2
= –3x
2x
Use algebra:
16 xy
= 4y
4x
2. a) 3k – 1 b) 9k 2 divided by 3 is 3k 2 and 3k divided by 3 is k.
Practise
3. a)
6x2 + 4x
4x2 + 6x
= 3x + 2 b)
= 2x + 3
2x
2x
4. a)
6 x 2 − 3x
− x2 − 4x
= 2x – 1 b)
= –x – 4
3x
x
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MHR ● Chapter 7: Multiplying and Dividing Polynomials
Multiplying Polynomials by Monomials:
10. a) y b) 2.1r
Draw a rectangle: (2x)(3x + 1)
11. 6y2 + 10y
12. a) (3x)(2x + 4) = 6x2 + 12x b) (2x)(–3x – 1) = –6x2 – 2x
6x2
3x
2x
13. a) 40x2 – 20x b) –3.6x2 – 18x
2x
14. a) (x2 + 5x) ÷ x = x + 5 b) (– 4x2 + 12x) ÷ (–2x) = 2x – 6
15. a) 6n – 1 b) 5 – x
3x + 1
1
16. The height of the triangle is 2x + 4.
Use algebra tiles: (2x)(3x + 1) = 6x2 + 2x
Key Word Builder, page 401
a) polynomial b) distributive property c) binomial d) monomial
L: 9x2 – 9x I: 6xy O: 4x – 1 A: 2x2 N: 18 M: 5
Mae’s answer was a monomial when she divided.
Chapter 7 Practice Test, pages 402–404
1. B 2. D 3. C 4. C 5. B 6. B
7. 5.2xy
8. –36h2 + 24h
9. x – 1
10. a) A = (4x)(3.1x) or A = 12.4x 2 b) The area of the patio that will need
paving stones is 8.9x2.
Use algebra: (2x)(3x + 1) = 6x2 + 2x
Dividing Polynomials by Monomials:
11. a) 2w + 15 b) (2w + 15)(w) = 2w2 + 15w
4 x2 − 6 x
Use algebra tiles:
= 2x − 3
2x
Math Link: Wrap It Up!, page 405
Answers will vary. Examples:
a) Beach volleyball court: length = 18 m; width = 9 m
Wading pool: radius = 5 m
b) Beach volleyball court: 0.5 m; Wading pool: 0.4 m
d) Beach volleyball court: A = 2x2 – 5x; Wading pool: A = 28.26y2
e) Beach volleyball court: V = x2 – 2.5x; Wading pool: 11.3y2
Use algebra:
Challenge, page 406
4 x2 − 6 x
= 2x – 3
2x
Answers will vary.
Chapters 5–7 Review, pages 407–411
Chapter 7 Review, pages 397–400
1. 4y 2. 4x – 2 3. 4x2 – 2x 4. 4xy – 2x 5. 4xy 6. 2x2 – 2x
7. a) (2x)(4x) = 8x2
1. a) –1
b) (–3x)(3x) = –9x2
b) –3g2 + g + 5
8. a) – 88x2 b) 5.5x2
9. a)
6x2
= 3x
2x
b)
− 8x2
= –2x
4x
2. a) 7m – 2 b) 4w2 + w + 4 c) –3z – 2 d) 2d 2 – 5
3. a) n + 4 b) 10 c) 10(n + 4) d) 10n + 40
4. a)
Type of income
Variable
Comic book
c
Hardcover Book
h
Paperback novel
p
Expression
10c
8h
3p
b) I = 10c + 8h + 3p
Answers ● MHR 417
5. a) 4, 4
b)
Figure Number, n
7. a) $500 b) 3 h
Number of Tiles, t
8. a) 12x2 b) –5y2
1
4
9. a) – 6h b) 8.4x
2
8
3
12
4
16
c) t = 4n d) 32 tiles
6. a)
Week, w
0
1
2
3
4
5
10. a) 18x2 – 15x b) 3w2 + 1.5w
11. a) Length = 2w + 3 b) The expression is 2w2 + 3w.
12. a) 3g + 2 b) –2x + y
13. The expression for the width is 2x – 1.
Task, pages 412–413
Amount in the
Bank, A ($)
112
137
162
187
212
237
1. a) The best size of TV is 110 cm. b) Answers will vary. c) Answers
will vary. Example: 280 cm d) A 94 cm TV would be the best.
2. 8.3 m2
3. Answers will vary. Example: NO. It is too big for the classroom.
b) A = 25w + 112 c) It will take her 14 weeks.
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MHR ● Chapter 7: Multiplying and Dividing Polynomials