Name: ___________________________________
Get Ready
Classify Polynomials
1. Classify each polynomial by the number of
terms.
a) −2y
b) x2 + 3x + 2
c) 6x2y + 2xy + 4
d) x2 + y2
e) 3x2 + 2x + y − 4
Date: _______________________________
…BLM 5–1...
6. Expand using the distributive property.
a) 6m(2m − 4)
b) −8xy(2x − y)
c) 6a2(−3a + 4ab)
d) −2a(b2 − 6ab + 7)
7. A rectangular prism has the dimensions
shown.
2. State the degree of each polynomial.
a) x2 + 3x − 1
b) x + 2y + 4z
c) 6 + 2y3 + xy
d) 7a3b2 + 6a2b2 − 7ab
Add and Subtract Polynomials
3. Simplify.
a) (3x + 7) + (3x − 6)
b) (2a − b) + (6a − 4b)
c) (3x2 + 2x − 4) − (2x2 − 5x + 1)
d) (9y3 − 7y2 + 4) − (3y3 + 2y2 − 1)
4. Simplify.
a) (6x2 + 2xy − 3y2) + (8x2 − 4xy − 2y2)
b) (8ab2 + 8a − b2) − (9ab2 − 7a + b2)
c) (6x − 8) − (4x + 7) + (6x − 2)
d) (6a2 + b) − (2b − 3a2) − (11b2 + 9a2 + 2)
The Product of a Monomial and a Polynomial
5. Expand using the distributive property.
a) 2x(x + y)
b) −8(6a2 − 4a)
c) −6(a + 7)
d) 2(3x2 + 2x + 4)
Principles of Mathematics 10: Teacher’s Resource
BLM 5–1 Get Ready
a) Find a simplified expression for the volume.
b) Find a simplified expression for the surface
area.
Factors
8. Write all of the factors of each number.
a) 6
b) 34
c) 17
d) 44
9. Write each number as the product of prime
factors.
a) 12
b) 9
c) 40
d) 55
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
Section 5.1 Practice Master
1. What binomial product does each model
represent?
a)
b)
…BLM 5–3...
5. Expand and simplify.
a) 2(x − 7)(2x + 1)
b) (x + 3)(x + 6) − 2(x + 1)
c) −(x − 4)(x − 1) + 5(3x − 1)(2x + 1)
d) (m − 2)2 − (3m + 2)2
e) −(m + 7)(m − 1) + 4(2m + 1)(3m − 4)
f) −6(2x + 1)(6x + 1) + 3(4x − 3)2
6. Write and simplify an expression to represent
the area of each shaded region.
a)
b)
2. Model each product using algebra tiles, virtual
tiles, or a diagram.
a) 3x(x + 3)
b) (x + 3)(x + 2)
c) (x + 1)(2x + 1)
3. Use the distributive property to find each
binomial product.
a) (x − 2)(x + 3)
b) (y + 6)(y + 2)
c) (n + 4)(n − 5)
d) (d + 6)(d + 7)
e) (x − 8)(x − 6)
f) (a − 6)(a − 3)
4. Use the distributive property to find each
binomial product.
a) (x − 2y)(x + 2y)
b) (2x + 1)(x − 3)
c) (k − 6)(k + 7)
d) (2p − 7q)(2p − 5q)
e) (3 − 2s)(2 − 3s)
f) (−2t − r)(−3t + r)
Principles of Mathematics 10: Teacher’s Resource
BLM 5–3 Section 5.1 Practice Master
7. A rectangular prism has a width of
x centimetres. Its length is 4 cm more than its
width and its height is 5 cm more than its
width.
a) Draw a diagram of the prism.
b) Write a simplified expression for the
volume of the prism.
c) Write a simplified expression for the
surface area of the prism.
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
…BLM 5–4...
Section 5.2 Practice Master
1. Draw a diagram to represent each product.
a) (x + 3)2
b) (x + 2)2
2. Expand and simplify.
a) (x + 4)2
b) (y + 7)2
c) (a + 8)2
d) (q + 5)2
3. Expand and simplify.
a) (3y + 6)2
b) (3x + 2y)2
c) (2x + y)2
d) (6c + 7d)2
4. Expand and simplify.
a) (x − 6)2
b) (b − 25)2
c) (r − 11)2
d) (e − 7)2
5. Expand and simplify.
a) (8a − 1)2
b) (2u − 3v)2
c) (6p − 7)2
d) (5q − 8r)2
6. Expand and simplify.
a) (v − 2)(v + 2)
b) (x + 6)(x − 6)
c) (x + y)(x − y)
d) (r − s)(r + s)
Principles of Mathematics 10: Teacher’s Resource
BLM 5–4 Section 5.2 Practice Master
7. Expand and simplify.
a) (6g − 7h)(6g + 7h)
b) (3x + y)(3x − y)
c) (g − 9x)(g + 9x)
d) (4x − 5y)(4x + 5y)
8. A cube has length, width, and height of
x metres. Each dimension is increased by
y metres.
a) Write a simplified formula for the volume
of the new cube.
b) Write a simplified formula for the surface
area of the new cube.
9. A parabola has equation y = (x − 3)2.
a) Identify the coordinates of the vertex.
b) Expand and simplify the equation.
c) Verify that the coordinates of the vertex
satisfy your equation from part b).
10. The side length of a square is represented by
x centimetres. The length of a rectangle is
3 cm greater than the side length of the
square. The width of the rectangle is 3 cm
less than the side length of the square. Which
figure has the greater area and by how much?
11. Expand and simplify.
a) (4x2 + 3y2)2
b) (3x2 + 2y2)(3x2 − 2y2)
c) (x − 3)2 − (x + 3)(x − 3)
d) 3(2b + 1)(2b − 1) + (b − 3)2
e) (3x2 + 5x − 1)2
f) (2x − 3)3
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
…BLM 5–6...
Section 5.3 Practice Master
1. Use algebra tiles or a diagram to illustrate
the factoring of each polynomial.
a) x2 + 3x
b) 2x2 + 10x
c) 3x2 + 6x
2. Factor fully.
a) 3x + 6y
b) 17ac − 34ad
c) 16x2y2 − 24xy
d) 27x3y3 + 18x2y2 + 9xy
e) 6n2p2 + 12np2 + 36n3p3
f) 33c4d3e2 − 11c2de
g) 3g2 + 6g + 9
3. Factor fully.
a) 2x(x + 7) + 3(x + 7)
b) a(b − 7) + 2(b − 7)
c) 4s(r + u) − 3(r + u)
d) y(x + s) + z(x + s)
6. Factor.
a) 3x(6 − y) + 2(y − 6)
b) 2y(x − 3) + 4z(3 − x)
7. Write an expression in factored form for the
area of each shaded region.
a)
b)
4. Factor by grouping.
a) ax + ay + 3x + 3y
b) 4x2 + 6xy + 12y + 8x
c) y2 + 3y + ay + 3a
d) 25x2 + 5x + 15xy + 3y
5. The formula for the surface area of a
rectangular prism is SA = 2lw + 2lh + 2wh.
a) Write this formula in factored form.
b) If l is 10 cm, w is 5 cm, and h is 8 cm, find
the surface area using both the original
formula and the factored form. What do
you notice? Explain why this is so.
Principles of Mathematics 10: Teacher’s Resource
BLM 5–6 Section 5.3 Practice Master
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
Section 5.4 Practice Master
…BLM 5–7...
1. Illustrate the factoring of each trinomial using
algebra tiles or a diagram.
a) x2 + 5x + 6
b) x2 + 6x + 9
c) x2 + 8x + 15
d) x2 + 12x + 27
5. Determine two values of b so that each
expression can be factored.
a) x2 + bx + 12
b) x2 − bx + 18
c) x2 + bx − 15
d) x2 − bx − 18
2. Find two integers with the given product and
sum.
a) product = 48 and sum = 14
b) product = −15 and sum = 2
c) product = −30 and sum = −1
d) product = 2 and sum = −3
6. Determine two values of c so that each
expression can be factored.
a) x2 + 4x + c
b) x2 − 9x + c
3. Factor, if possible.
a) x2 + 8x + 12
b) c2 − 3c − 18
c) d2 + 10d + 21
d) d2 − 12d + 35
e) x2 + x + 1
f) c2 − 11c + 30
g) y2 + 15y + 56
h) x2 − x − 72
7. A parabola has equation y = 3x2 − 30x + 48.
a) Factor the right side of the equation fully.
b) Identify the x-intercepts of the parabola.
c) Find the equation of the axis of symmetry,
find the vertex, and draw a graph of the
parabola.
8. Determine expressions to represent the
dimensions of this rectangular prism.
4. Factor fully by first removing the greatest
common factor (GCF).
a) 3x2 − 12x − 36
b) −2x2 + 2x + 4
c) 6x2 − 42x + 72
d) −3x2 − 18x − 24
e) 4x2 − 40x + 84
f) x3 + 7x2 + 12x
Principles of Mathematics 10: Teacher’s Resource
BLM 5–7 Section 5.4 Practice Master
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
…BLM 5–8...
Section 5.5 Practice Master
1. Use algebra tiles or a diagram to factor each
trinomial.
a) 2x2 + 7x + 3
b) 6x2 + 11x + 4
c) 3x2 + 7x + 2
d) 4x2 + 18x + 20
5. The area of a rectangular parking lot is
represented by A = 6x2 − 19x − 7.
a) Factor the expression to find expressions
for the length and width.
b) If x represents 15 m, what are the length
and width of the parking lot?
2. Factor.
a) 6x2 + 10x − 4
b) 56x2 − 9x − 2
c) 9x2 + 6x + 1
d) 12c2 − 26c − 16
e) 2d2 − 11d − 6
f) 2r2 + 13r + 20
g) 6s2 − 29s + 35
h) 15r2 − 7r − 2
i) 4r2 − 20r + 25
j) 13x2 − 57x + 20
6. The height, h, in metres, of a baseball above
the ground relative to the horizontal distance,
d, in metres, from the batter is given by
h = −0.005d2 + 0.49d + 1.
a) Write the right side of the equation in
factored form. Hint: First divide each term
by the common factor, −0.005.
b) At what horizontal distance from the batter
will the baseball hit the ground if it is not
caught by an outfielder?
3. Factor.
a) 6x2 − 5xy − 4y2
b) 9x2 + 12xy + 4y2
c) 12r2 + 7rs − 10s2
d) 15r2 − 23rs + 4s2
e) 2x2 − 19xy + 42y2
f) 18y2 + 21yx − 4x2
4. Find two values of k so that each trinomial
can be factored over the integers.
a) 12x2 + kx + 14
b) 6x2 + kx + 10
c) 4x2 − 12x + k
d) kx2 − 40xy + 16y2
Principles of Mathematics 10: Teacher’s Resource
BLM 5–8 Section 5.5 Practice Master
7. Sydney Harbour Bridge in Australia is
unusually wide for a long-span bridge.
It carries two rail lines, eight road lanes,
a cycle lane, and a walkway.
a) Factor the expression 10x2 − 7x − 3 to find
binomials that represent the length and the
width of the bridge.
b) If x represents 50 m, what are the length
and the width of the bridge, in metres?
8. Factor.
a) 10x4 − 3x2 − 18
b) 20x6 − 59x3y2 + 42y4
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
…BLM 5–10...
Section 5.6 Practice Master
1. Factor.
a) x2 − 25
b) y2 − 49
c) 9k2 − 1
d) 16k2 − 49
e) 25w2 − 36
f) 4 − 9w2
2. Factor.
a) x2 − y2
b) 36x2 − y2
c) 25r2 − 36s2
d) 144r2 − 49s2
e) 121x2 − 9y2
f) 100r2 − 81s2
3. Factor.
a) x2 + 14x + 49
b) x2 − 6x + 9
c) x2 − 8x + 16
d) 100 − 20x + x2
e) 4x2 − 12xy + 9y2
f) 49x2 + 56xy + 16y2
5. Determine the value(s) of b so that each
trinomial is a perfect square.
a) bx2 + 10xy + y2
b) 36x2 − bxy + 49y2
6. Determine two values of k so that each
trinomial can be factored as a difference of
squares.
a) 25x2 − ky2
b) kx2 − 16
7. Factor, if possible.
a) (5c + 3)2 − (2c + 1)2
b) 100 + (x − 3)2
c) 9x2 + 8x + 25
d) 25x2y2 − 150xyab + 225a2b2
8. A parabola has equation y = 4x2 + 32x + 64.
Rewrite the equation in factored form to find
the coordinates of the vertex.
9. Find an algebraic expression for the area of
the shaded region in factored form.
4. Factor fully, if possible.
a) 2a2 + 12a + 18
b) 25x2 − 16y
c) 75x2 + 210xy + 147y2
d) 9x3y − 16xy3
e) 36m2 − 96mn + 64n2
f) 20x2 + 20xy + 5y2
Principles of Mathematics 10: Teacher’s Resource
BLM 5–10 Section 5.6 Practice Master
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
…BLM 5–11...
(page 1)
Chapter 5 Review
5.1 Multiply Polynomials
1. Use the distributive property to find each
binomial product.
a) (x + 7)(x + 3)
b) (y − 3)(y + 5)
c) (x − 3y)(x + 2y) d) (3a + 8b)(5a + 6b)
2. Expand and simplify.
a) −4(a + 6)(a − 3)
b) −3x(x + 2y)(x + 6y)
c) (10y + 6)(3y + 7) − (y + 2)(y − 4)
d) 2b(4b − 7)(3b + 2) − b(5b + 2)(b − 6)
e) −x(x + y)(2x + y) − y(3x + y)(x − y)
3. A parabola has equation y = 2(x − 3)(x − 6).
a) Expand and simplify the right side of
the equation.
b) State the x-intercepts of the parabola.
c) Verify in the expanded form that these
are the x-intercepts.
4. a) Write a simplified algebraic expression
to represent the area of the figure.
8. Expand and simplify.
a) (x − 3y)2
b) −5(2x + 5b)2
c) (11x − 13y)(11x + 13y)
d) −(a − 6b)(a + 6b)
9. A square has side length 4a. One dimension
is increased by 6 and the other is decreased
by 6.
a) Write an algebraic expression to represent
the area of the resulting rectangle.
b) Expand this expression and simplify.
c) Write and simplify an algebraic expression
for the change in area from the square to
the rectangle.
d) Calculate the new area of the rectangle
if a represents 5 cm.
5.3 Common Factors
10. Use algebra tiles or a diagram to illustrate
the factoring of each polynomial.
a) x2 + 5x
b) 8x2 + x
11. Factor.
a) 2x2 + 4x
c) 10x2 + 20y2
b) Expand and simplify your expression
from part a).
5.2 Special Products
5. Draw a diagram to illustrate each product.
a) (x + 5)2
b) (y + 3)2
6. Expand and simplify.
b) (r − 3)2
a) (x + 6)2
c) (y + 10)2
d) (e − 5)2
b) 5x2 + 3x
d) 3xy − 7xz
12. Factor by grouping.
a) 2x2 + 2x + 3xy + 3y
b) x3 + x2y + yx + y2
c) 5ab − 5a + 3b − 3
d) 3a2x + 3a2y + b2x + b2y
13. Factor, if possible.
a) 2z(x + y) + 3xy(x + y)
b) x2 + y2 + z2
c) 6a3 + 3a2 + 12a + 6
d) x2yz2 − x2z2 + xyz
7. Expand and simplify.
a) (b + 9)(b − 9)
b) (y − 11)(y + 11)
c) (m + 13)(m − 13) d) (14 − x)(14 + x)
Principles of Mathematics 10: Teacher’s Resource
BLM 5–11 Chapter 5 Review
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
…BLM 5–11...
(page 2)
14. Write an expression in fully factored form
for the shaded area.
5.4 Factor Quadratic Expressions of the Form
x2 + bx + c
15. Illustrate the factoring of each trinomial
using algebra tiles or a diagram.
a) x2 + 6x + 9
b) x2 + 12x + 35
16. Factor.
a) x2 − 4x − 12
b) x2 − 7x + 12
c) x2 − 4x − 45
d) x2 + 9x + 14
17. Factor completely by first removing the
greatest common factor (GCF).
a) −2x2 + 16x − 30
b) x3 + 3x2 − 28x
18. Determine binomials to represent the length
and width of the rectangle, and then
determine the dimensions of the rectangle if
x = 11 cm.
5.5 Factor Quadratic Expressions of the Form
ax2 + bx + c
19. Factor, using algebra tiles or a diagram if
necessary.
a) 12x2 − 5x − 3
b) 3x2 − 13x − 10
c) 10x2 + 9x − 7
d) 21x2 + 4x − 1
20. Factor, if possible.
a) 3x2 + 15y + 33
b) 2x2 + 7x + 9
c) 30x2 + 9x − 12
d) −6x2 − 34x + 12
21. Find a value of k so that each trinomial can
be factored over the integers.
a) 3x2 + kx − 10
b) 24x2 + 47x + k
5.6 Factor a Perfect Square Trinomial and
a Difference of Squares
22. Factor fully.
a) x2 − 100
b) c2 − 25
c) 9x2 − 16
d) 128 − 18x2
e) 1 − 225y2
f) −3x2 + 27y2
23. Verify that each trinomial is a perfect square,
and then factor.
a) y2 + 16y + 64 b) x2 − 20x + 100
c) 225 − 90y + 9y2 d) 121c2 + 308cd + 196d2
24. Factor, if possible.
b) 50x2 − 60xy + 18y2
a) 9y2 + 24y − 16
c) (x − 3)2 − (y − 4)2 d) x2 + 9y2
25. A rectangular prism has a volume of
4x3 + 12x2 + 9x.
a) Determine algebraic expressions for the
dimensions of the prism.
b) Describe the faces of the prism.
c) Determine the volume if x = 3 cm.
d) Determine the surface area if x = 3 cm.
Principles of Mathematics 10: Teacher’s Resource
BLM 5–11 Chapter 5 Review
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
…BLM 5–13...
(page 1)
Chapter 5 Practice Test
1. What binomial product does each diagram
represent?
a)
4. If it is possible to remove a common factor
from the expression 2x2 + ky + 4, where k is
an integer, what can you state about the
possible values of k? Explain.
5. a) Write an algebraic expression for the
volume of this rectangular prism.
b)
b) Expand and simplify the expression.
c) Find the volume if x = 2.
6. Factor fully.
a) x2 + 10x + 25
b) 25r2 − 20rs + 4s2
c) 5x2 − 5
d) 1 − 49m2
e) 5m2 + 17m + 6
f) m2 − 9mn + 14n2
2. Expand and simplify.
a) −2x3(4x2 + 2x + 4)
b) −xy(6x2 + xy + 1) − 2(x3y + 4xy)
3. Expand and simplify.
a) (x − 3)(x − 9)
b) (2x + 3)(2x − 1)
c) −3(x − 4)2 + 2(x − 3)(x + 3)
d) (3c + d)2 + 2c(c − d)
e) 2(x − 1)(x − 6) − 3(2x − 1)2
f) (2c + 3d)2 − 3(c + 1)2
Principles of Mathematics 10: Teacher’s Resource
BLM 5–13 Chapter 5 Practice Test
7. Factor, if possible.
a) 3y3 − 7y2 + 2y
b) 4m2 + 16
c) 6y2 + y − 1
d) x(m − 2) − 4(m − 2)
e) y2 + 2x + 2y + xy
f) 9t − 4t3
8. A ball is thrown into the air and its path is
given by h = −5t2 + 20t + 25, where h is the
height, in metres, above the ground and t is
the time, in seconds.
a) Factor the right side of the equation fully.
b) When does the ball hit the ground?
c) Find the height of the ball 2 s after it is
thrown.
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
…BLM 5–13...
(page 2)
9. Determine two values of k so that each
expression can be factored over the integers.
a) x2 + kx + 36
b) 3x2 − 8x + k
c) 36x2 − kxy + 49y2
d) 49x2 − ky2
10. Write and simplify an algebraic expression
for the area of the shaded region.
11. The volume of a rectangular prism is
represented by the equation V = 12x3 − 3x.
a) Factor the right side of the equation fully.
b) Draw a diagram of the prism.
c) If x represents 6 cm, what are the
dimensions of the prism?
13. Describe the steps needed to determine
whether the expression ax3 + bx2 + cx can
be factored over the integers.
14. Factor to evaluate each difference.
a) 232 − 222
b) 252 − 232
c) 812 − 772
d) 1542 − 1502
15. a) Two numbers that differ by 2 can be
multiplied by squaring their average and
then subtracting 1. For example,
14 × 16 = 152 − 1, which is 225 − 1,
or 224. How does the product of the sum
and difference (x − 1)(x + 1) explain the
method?
b) Develop a similar method for multiplying
two numbers that differ by 4.
c) Show how the product of a sum and
a difference explains your method from
part b).
12. The face of a Canadian $20 bill has an area
that can be represented by the expression
10x2 + 9x − 40.
a) Factor 10x2 + 9x − 40 to find expressions
to represent the dimensions of the bill.
b) If x represents 32 mm, what are the
dimensions of the bill, in millimetres?
Principles of Mathematics 10: Teacher’s Resource
BLM 5–13 Chapter 5 Practice Test
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
…BLM 5–14...
(page 1)
Chapter 5 Test
1. Represent each binomial product using
a diagram.
a) (x + 3)(2x + 1)
b) (2x + 3)(3x + 2)
2. Expand and simplify.
a) 3x2y(y2x + 4xy − 2y2)
b) −4(x2 + 3x − 11) + 5x(x − 4)
6. Factor.
a) x2 − 10x + 25
b) 4x2 − 12x + 9
c) 2y2 + 5y + 2
d) 3k2 − 11k − 4
e) 10r2 + r − 3
f) 6s2 − 11st − 10t2
3. Expand and simplify.
a) (k + 4)(k − 1)
b) (6x − 1)(x − 5)
c) 2(3x − 2)2 − 3(x − 1)(x + 5)
d) (2x − 3y)2 + 2y(y − x)
e) 6(1 − x)(x + 4) − 2(5 − 2x)2
7. Factor fully, if possible.
a) 21x2 + 21x − 42
b) 7g2 + 28g − 147
c) 3x2 + 11x − 13
d) c3 − 9c
e) 6d2 − 13d + 6
f) 50x2 − 72
4. In baseball, the first base bag is a square.
Its side length can be represented by the
expression 5x + 3.
a) Write and expand an expression to
represent the area of the top of the bag.
b) If x represents 7 cm, what is the area of
the top of the bag, in square centimetres?
8. A stone is thrown straight down from a tall
building. The relation h = 150 − 5t − 5t2
approximates its path, with h in metres and
t in seconds.
a) How tall is the building?
b) Factor the right side of the equation fully.
c) When does the stone hit the ground?
5. a) Write an algebraic expression for the
volume of the rectangular prism.
9. The North Stone Pyramid at Dahshur in
Egypt has a square base with an area that
can be represented by the trinomial
9x2 − 12x + 4.
a) Factor the trinomial to find a binomial to
represent the side length of the base of the
pyramid.
b) If x represents 74 m, what is the side
length of the base, in metres?
b) Expand and simplify the expression.
c) Find the volume if x = 1 cm.
Principles of Mathematics 10: Teacher’s Resource
BLM 5–14 Chapter 5 Test
Copyright © 2007 McGraw-Hill Ryerson Limited
Name: ___________________________________
Date: _______________________________
…BLM 5–14...
(page 2)
10. Determine the value(s) of k so that each
trinomial can be factored as a perfect square.
a) x2 − 12x + k
b) 9x2 + kx + 25
c) kx2 − 4x + 1
d) 16y2 + ky + 9
e) x2 + kx + 36
f) 4x2 − 24x + k
g) 36x2 − kxy + 49y2
h) 49x2 − 42xy + ky2
13. Explain how to determine the value(s) of k
that would make x2 + kx + 100 a perfect
square trinomial.
14. If a and b are integers, find values of a and b
so that a2 – b2 is 21.
15. a) Write an expression for the shaded area in
the diagram.
11. Write and simplify an algebraic expression
for the area of the shaded region.
12. The area of a rectangle is represented by the
equation A = 6x2 + 5x − 4.
a) Factor the right side of the equation fully
to find expressions for the length and
width of the rectangle.
b) Find an expression for the perimeter of
the rectangle.
c) If x represents 7 cm, find the area and the
perimeter of the rectangle.
Principles of Mathematics 10: Teacher’s Resource
BLM 5–14 Chapter 5 Test
b) Factor this expression.
c) Find the area of the shaded region if
x = 3 cm.
Copyright © 2007 McGraw-Hill Ryerson Limited
…BLM 5–16...
(page 1)
BLM Answers
Get Ready
1. a)
c)
e)
2. a)
3. a)
c)
4. a)
c)
5. a)
c)
6. a)
c)
7. a)
8. a)
c)
9. a)
c)
monomial b) trinomial
trinomial
d) binomial
four-term polynomial
2 b) 1 c) 3 d) 5
6x + 1
b) 8a − 5b
x2 + 7x − 5
d) 6y3 − 9y2 + 5
b) −ab2 + 15a − 2b2
14x2 − 2xy − 5y2
8x − 17
d) −11b2 − b − 2
2
2x + 2xy
b) −48a2 + 32a
−6a − 42
d) 6x2 + 4x + 8
2
b) −16x2y + 8xy2
12m − 24m
3
3
−18a − 24a b
d) −2ab2 + 12a2b − 14a
3
2
30x + 20x
b) 62x2 + 28x
1, 2, 3, 6
b) 1, 2, 17, 34
1, 17
d) 1, 2, 4, 11, 22, 44
2×2×3
b) 3 × 3
2×2×2×5
d) 5 × 11
b)
c)
Section 5.1 Practice Master
1. a) (x + 1)(x + 3) b) (x + 2)(x + 2)
2. Diagrams may vary. For example:
a)
3. a)
c)
e)
4. a)
c)
e)
5. a)
c)
e)
6. a)
b)
7. a)
x2 + x − 6
b) y2 + 8y + 12
2
d) d2 + 13d + 42
n − n − 20
2
f) a2 − 9a + 18
x − 14x + 48
2
2
b) 2x2 − 5x − 3
x − 4y
2
d) 4p2 − 24pq + 35q2
k + k − 42
2
6 − 13s + 6s
f) 6t2 + rt − r2
2
b) x2 + 7x + 16
4x − 26x − 14
2
d) −8m2 − 16m
29x + 10x − 9
2
f) −24x2 − 120x + 21
23m − 26m − 9
2x(3x + 1) + 2x(2x + 2 − 2x) = 6x2 + 6x
(x + 6)(x − 3) − 2x(x + 1) = −x2 + x − 18
b) x3 + 9x2 + 20x
Principles of Mathematics 10: Teacher’s Resource
Chapter 5 Practice Masters Answers
c) 6x2 + 36x + 40
Copyright © 2007 McGraw-Hill Ryerson Limited
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Section 5.2 Practice Master
1. Diagrams may vary. For example:
a)
11. a)
c)
e)
f)
16x4 + 24x2y2 + 9y4 b) 9x4 − 4y4
−6x + 18
d) 13b2 − 6b + 6
9x4 + 30x3 − 19x2 − 10x + 1
8x3 − 36x2 + 54x − 27
(page 2)
Section 5.3 Practice Master
1. Diagrams may vary. For example:
a)
b)
b)
2. a) x2 + 8x + 16
b) y2 + 14y + 49
2
c) a + 16a + 64
d) q2 + 10q + 25
2
3. a) 6y + 36y + 36 b) 6x2 + 12xy + 4y2
c) 4x2 + 4xy + y2
d) 36c2 + 84cd + 49d2
2
b) b2 − 50b + 625
4. a) x − 12x + 36
c) r2 − 22r + 121 d) e2 − 14e + 49
5. a) 64a2 − 16a + 1 b) 4u2 − 12uv + 9v2
c) 36p2 − 84p + 49 d) 25q2 − 80qr + 64r2
6. a) v2 − 4
b) x2 − 36
2
2
d) r2 − s2
c) x − y
2
2
b) 9x2 − y2
7. a) 36g − 49h
2
2
c) g − 81x
d) 16x2 − 25y2
3
2
8. a) V = x + 3x y + 3xy2 + y3
b) SA = 6x2 + 12xy + 6y2
9. a) (3, 0)
b) y = x2 − 6x + 9
c) Substitute the coordinates into the left
and right sides of the equation.
L.S. = x 2 − 6 x + 9 R.S. = y
=0
= 32 − 6(3) + 9
c)
= 9 − 18 + 9
=0
L.S. = R.S.
10. the square, by 9 cm2
Principles of Mathematics 10: Teacher’s Resource
Chapter 5 Practice Masters Answers
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…BLM 5–16...
(page 3)
3(x + 2y)
b) 17a(c − 2d)
8xy(2xy − 3)
d) 9xy(3x2y2 + 2xy + 1)
6np2(n + 2 + 6n2p) f) 11c2de(3c2d2e − 1)
3(g2 + 2g + 3)
(x + 7)(2x + 3)
b) (b − 7)(a + 2)
(r + u)(4s − 3)
d) (x + s)(y + z)
(a + 3)(x + y)
b) 2(x + 2)(2x + 3y)
(y + 3)(y + a)
d) (5x + 1)(5x + 3y)
SA = 2(lw + lh + wh)
Both formulas give 340 cm2 because the formulas
are equivalent.
6. a) −(y − 6)(3x − 2) b) 2(x − 3)(y − 2z)
5
b) r2(π − 2)
7. a) xy (9 x − 2)
2
2. a)
c)
e)
g)
3. a)
c)
4. a)
c)
5. a)
b)
Section 5.4 Practice Master
c)
d)
1. Diagrams may vary. For example:
a)
b)
2. a) 6, 8 b) −3, 5 c) −6, 5 d) −2, −1
3. a) (x + 6)(x + 2)
b) (c − 6)(c + 3)
c) (d + 3)(d + 7)
d) (d − 5)(d − 7)
e) not possible
f) (c − 5)(c − 6)
g) (y + 7)(y + 8)
h) (x − 9)(x + 8)
4. a) 3(x − 6)(x + 2)
b) −2(x − 2)(x + 1)
c) 6(x − 4)(x − 3)
d) −3(x + 2)(x + 4)
e) 4(x − 7)(x − 3)
f) x(x + 3)(x + 4)
5. Answers may vary. For example:
a) b = 8, b = −8
b) b = 9, b = −9
c) b = 2, b = −2
d) b = 3, b = 7
6. Answers may vary. For example:
a) c = 3, c = −5
b) c = −10, c = 8
7. a) 3(x − 8)(x − 2)
b) 2, 8
c) x = 5, (5, −27)
8. x by x + 2 by x + 3
Principles of Mathematics 10: Teacher’s Resource
Chapter 5 Practice Masters Answers
Copyright © 2007 McGraw-Hill Ryerson Limited
…BLM 5–16...
(page 4)
Section 5.5 Practice Master
1. a) (x + 3)(2x + 1)
b) (2x + 1)(3x + 4)
c) (x + 2)(3x + 1)
d) 2(x + 2)(2x + 5)
2. a) 2(x + 2)(3x − 1) b) (7x − 2)(8x + 1)
c) (3x + 1)(3x + 1) d) 2(2c + 1)(3c − 8)
e) (d − 6)(2d + 1) f) (r + 4)(2r + 5)
g) (2s − 5)(3s − 7) h) (3r − 2)(5r + 1)
i) (2r − 5)(2r − 5) j) (x − 4)(13x − 5)
3. a) (2x + y)(3x − 4y) b) (3x + 2y)(3x + 2y)
c) (3r − 2s)(4r + 5s) d) (3r − 4s)(5r − s)
e) (x − 6y)(2x − 7y) f) (3y + 4x)(6y − x)
4. Answers may vary. For example:
a) k = 26, k = −29 b) k = 16, k = −17
c) k = 5, k = −7
d) k = 9, k = −11
5. a) length 3x + 1, width 2x − 7
b) length 46 m, width 23 m
6. a) −0.005(x − 100)(x + 2) b) 100 m
7. a) 10x + 3, x – 1
b) 53 m; 49 m
8. a) (2x2 − 3)(5x2 + 6)
b) (5x3 − 6y2)(4x3 − 7y2)
3. a) y = 2x2 − 18x + 36 b) 3, 6
c) Substitute the coordinates of the points at the
x-intercepts into both sides of the equation.
L.S. = y R.S. = 2 x 2 − 18 x + 36
=0
= 2(6) 2 − 18(6) + 36
= 72 − 108 + 36
=0
L.S. = R.S.
4. a) (x − 3)(x + 9) − 9(x − 5) b) x2 − 3x + 18
5. Diagrams may vary. For example:
a)
Section 5.6 Practice Master
1. a) (x − 5)(x + 5)
b) (y − 7)(y + 7)
c) (3k − 1)(3k + 1)
d) (4k − 7)(4k + 7)
e) (5w − 6)(5w + 6)
f) (2 − 3w)(2 + 3w)
2. a) (x − y)(x + y)
b) (6x − y)(6x + y)
c) (5r − 6s)(5r + 6s)
d) (12r − 7s)(12r + 7s)
e) (11x − 3y)(11x + 3y) f) (10r − 9s)(10r + 9s)
b) (x − 3)2
3. a) (x + 7)2
d) (10 − x)2
c) (x − 4)2
2
f) (7x + 4y)2
e) (2x − 3y)
2
4. a) 2(a + 3)
b) not possible
c) 3(5x + 7y)2
d) xy(3x − 4y)(3x + 4y)
f) 5(2x + y)2
e) 4(3m − 4n)2
5. a) b = 25
b) b = 84 or b = −84
6. Answers may vary, but k must be a perfect square.
For example:
a) k = 25, k = 4
b) k = 9, k = 81
7. a) (3c + 2)(7c + 4)2
b) not possible
c) not possible
d) 25(xy − 3ab)2
8. y = 4(x + 4)2; (−4, 0)
9. A = 4(x + 3)(2x + 1)
Chapter 5 Review
1. a)
c)
2. a)
c)
e)
x2 + 10x + 21
b) y2 + 2y − 15
2
2
d) 15a2 + 58ab + 48b2
x − xy − 6y
2
−4a − 12a + 72 b) −3x3 − 24x2y − 36xy2
29y2 + 90y + 50 d) 19b3 + 2b2 − 16b
−2x3 − 6x2y + xy2 + y3
Principles of Mathematics 10: Teacher’s Resource
Chapter 5 Practice Masters Answers
b)
6. a)
c)
7. a)
c)
8. a)
c)
9. a)
c)
x2 + 12x + 36
b) r2 − 6r + 9
2
y + 20y + 100 d) e2 − 10e + 25
b2 − 81
b) y2 − 121
2
d) 196 − x2
m − 169
2
2
b) −20x2 − 100xb − 125b2
x − 6xy + 9y
2
2
121x − 169y
d) 36b2 − a2
(4a + 6)(4a − 6)
b) 16a2 − 36
2
(4a)(4a) − (16a − 36) = 36 d) 364 cm2
Copyright © 2007 McGraw-Hill Ryerson Limited
…BLM 5–16...
(page 5)
10. Diagrams may vary. For example:
a)
b)
11. a) 2x(x + 2)
b) x(5x + 3)
d) x(3y − 7z)
c) 10(x2 + 2y2)
12. a) (x + 1)(2x + 3y)
b) (x + y)(x2 + y)
c) (5a + 3)(b − 1)
d) (3a2 + b2)(x + y)
13. a) (x + y)(2z + 3xy) b) not possible
c) 3(2a + 1)(a2 + 2) d) xz(xyz − xz + y)
14. xy(2x + 5)
15. Diagrams may vary. For example:
a)
b)
16. a) (x − 6)(x + 2)
b) (x − 3)(x − 4)
c) (x − 9)(x + 5)
d) (x + 2)(x + 7)
17. a) −2(x − 5)(x − 3)
b) x(x + 7)(x − 4)
18. length x − 9, width x − 10; length 2 cm, width 1 cm
19. a) (3x + 1)(4x − 3)
b) (x − 5)(3x + 2)
c) (2x − 1)(5x + 7)
d) (3x + 1)(7x − 1)
20. a) 3(x2 + 5y + 11)
b) not possible
c) 3(2x − 1)(5x + 4) d) −2(x + 6)(3x − 1)
21. Answers may vary. For example:
a) k = −7
b) k = 13
22. a) (x − 10)(x + 10)
b) (c − 5)(c + 5)
c) (3x − 4)(3x + 4)
d) 2(8 − 3x)(8 + 3x)
e) (1 − 15y)(1 + 15y)
f) −3(x − 3y)(x + 3y)
23. a) Since y2 = (y)2 and 64 = 82, the first and last terms
are perfect squares. Since 16y = 2(y)(8), the middle
term is twice the product of the square roots of the
first and last terms. Therefore, y2 + 16y + 64 is a
perfect square trinomial.
(y + 8)2
b) Since x2 = (x)2 and 100 = (−10)2, the first and last
terms are perfect squares. Since −20x = 2(x)(−10),
the middle term is twice the product of the square
roots of the first and last terms. Therefore,
x2 − 20x + 100 is a perfect square trinomial.
(x − 10)2
c) Since 225 = (15)2 and 9y2 = (−3y)2, the first and last
terms are perfect squares. Since −90y = 2(15)(−3y),
the middle term is twice the product of the square
roots of the first and last terms. Therefore,
225 − 90y + 9y2 is a perfect square trinomial.
(15 − 3y)2
This is not fully factored:
(15 − 3y)2 = [3(5 − y)]2 = 9(5 − y)2
d ) Since 121c2 = (11c)2 and 196d2 = (14d)2, the first
and last terms are perfect squares. Since
308cd = 2(11c)(14d), the middle term is twice the
product of the square roots of the first and last
terms. Therefore, 121c2 + 308cd + 196d2 is a perfect
square trinomial.
(11c + 14d)2
24. a) not possible
b) 2(5x − 3y)2
c) (x + y − 7)(x − y + 1) d) not possible
25. a) x by 2x + 3 by 2x + 3
b) two congruent squares and four congruent
rectangles
c) 243 cm3 d) 270 cm2
Chapter 5 Practice Test
1. a) (x + 3)(x + 2)
b) (2x + 1)(x + 4)
2. a) −8x5 − 4x4 − 8x3
b) −8x3y − x2y2 − 9xy
2
b) 4x2 + 4x − 3
3. a) x − 12x + 27
2
c) −x + 24x − 66
d) 11c2 + 4cd + d2
2
f) c2 + 12cd − 6c + 9d2 − 3
e) −10x − 2x + 9
4. k must be divisible by 2, because the only common
factor of the other two coefficients is 2.
Principles of Mathematics 10: Teacher’s Resource
Chapter 5 Practice Masters Answers
Copyright © 2007 McGraw-Hill Ryerson Limited
…BLM 5–16...
5. a) x2(3x + 1)(2x + 5)
b) 6x4 + 17x3 + 5x2
c) 252 cubic units
b) (5r − 2s)2
6. a) (x + 5)2
c) 5(x − 1)(x + 1)
d) (1 − 7m)(1 + 7m)
e) (m + 3)(5m + 2)
f) (m − 2n)(m − 7n)
7. a) y(y − 2)(3y − 1)
b) 4(m2 + 4)
c) (2y + 1)(3y − 1)
d) (m − 2)(x − 4)
e) (x + y)(y + 2)
f) t(3 − 2t)(3 + 2t)
8. a) −5(t − 5)(t + 1)
b) after 5 s
c) 45 m
9. Answers may vary. For example:
a) k = −12, k = 13 b) k = 5, k = 4 c) k = 84, k = 85
d) k = 1, k = 169 (k must be a perfect square)
10. π(r + 3)2 − πr2 = 3π(2r + 3)
11. a) 3x(2x − 1)(2x + 1)
b)
(page 6)
Chapter 5 Test
1. Diagrams may vary. For example:
a)
b)
c) 18 cm by 11 cm by 13 cm
12. a) (2x + 5)(5x – 8)
b) 69 mm by 152 mm
13. First, remove the greatest common factor, x, to get
x(ax2 + bx + c). Then, find two integers, m and n,
whose product is ac and whose sum is b. Break up the
middle term, bx, into mx + nx and then factor by
grouping.
14. a) 232 − 222 = (23 − 22)(23 + 22)
= 1(55)
= 55
b) 252 − 232 = (25 − 23)(25 + 23)
= 2(48)
= 96
c) 812 − 772 = (81 − 77)(81 + 77)
= 4(158)
= 632
d) 1542 − 1502 = (154 − 150)(154 + 150)
= 4(204)
= 816
15. a) You can write any two numbers that differ by 2 as
x – 1 and x + 1. Their average is
x −1+ x + 1 2x
=
or x. Their product is
2
2
2
(x – 1)(x + 1), or x – 1.
b) Find the square of their average and subtract 4.
c) Write the two numbers as x – 2 and x + 2. Their
average is x. Their product is (x – 2)(x + 2), or
x2 – 4.
Principles of Mathematics 10: Teacher’s Resource
Chapter 5 Practice Masters Answers
2. a) 3x3y3 + 12x3y2 − 6x2y3 b) x2 − 32x + 44
b) 6x2 − 31x + 5
3. a) k2 + 3k − 4
d) 4x2 − 14xy + 11y2
c) 15x2 − 36x + 23
2
e) −14x + 22x − 26
4. a) (5x + 3)2 = 25x2 + 30x + 9
b) 1444 cm2
5. a) 2x(5x − 2)(3x + 4)
b) 30x2 + 28x2 − 16x
c) 42 cm3
6. a) (x − 5)2
b) (2x − 3)2
c) (y + 2)(2y + 1)
d) (k − 4)(3k + 1)
e) (2r − 1)(5r + 3)
f) (2s − 5t)(3s + 2t)
7. a) 21(x − 1)(x + 2)
b) 7(g − 3)(g + 7)
c) not possible
d) c(c − 3)(c + 3)
e) (2d −3)(3d − 2)
f) 2(5x − 6)(5x + 6)
8. a) 150 m b) −5(t − 5)(t + 6) c) after 5 s
9. a) 3x – 2 b) 220 m
Copyright © 2007 McGraw-Hill Ryerson Limited
…BLM 5–16...
(page 7)
10. a) k = 36
b) k = −30, k = 30
c) k = 4
d) k = −24, k = 24
e) k = −12, k = 12 f) k = 36
g) k = −84, k = 84
h) k = 9
11. (2x + 3)(x + 4) − x2 = x2 + 11x + 12
12. a) (2x − 1)(3x + 4)
b) 10x + 6
c) 325 cm2, 76 cm
Principles of Mathematics 10: Teacher’s Resource
Chapter 5 Practice Masters Answers
13. Multiply the square root of 100, −10 or 10, by 2,
to get −20 or 20.
14. There are eight possible answers for the ordered pair
(a, b): (5, 2), (–5, 2), (5, –2), (–5, –2), (11, 10),
(–11, 10), (11, –10), and (–11, –10).
15. a) (2x + 3)(2x + 3) − (2)(2)
b) (2x + 1)(2x + 5)
c) 77 cm2
Copyright © 2007 McGraw-Hill Ryerson Limited