UNIT 9 - POLYNOMIALS AND EXPONENTS
ADDING/SUBTRACTING POLYNOMIALS (Day 1)
Polynomial: The sum of monomials.

Monomial:
____ term:
ex. ___________________________

Binomial:
____ unlike terms:
ex. ___________________________

Trinomial:
______ unlike terms:
ex. ___________________________
Simplest form/STANDARD FORM: When the polynomial contains no like terms and terms
are put in _________________________ exponent term to _______________ exponent term.
5x3 + 8x2 + 2x3 + 7
Coefficient: _______________________________________________________________________
DEGREE of a Polynomial: ___________________________________________________________
Constant Term: ____________________________________________________________________
Leading Coefficient: _______________________________________________________________
Simplest/Standard Form
3 x 2  x 5  x  7x
5x 3 2 x  7  8 x 3
2 x  3 x  9x 4
2
7 x  5 x  4  5x  4
a3  9a8  b  3b
8d6  5d  4e10  3  12g
1
Degree
Type of
Polynomial
Leading
Coefficient
Perform the operation indicated. Write answer in standard form.
1. (4x2 + x + 7) + (2x2 + 3x + 1)
2.
(3x3 – x2 + 8) – (x3 + 5x2 + 4x – 7)
3.
Find the difference between 10y and 3 y  6
4.
–5m + 6n + 8p – (6n + 3m)
5.
6.
Subtract 9a – 3b from 4a – 7b
7. From 2x3 – 4x2 + x, subtract 8x3 + 2x2 – 3x
8.
Describe and correct the error(s) in the following problems.
(a)
9.
(2x + 4x2 – 7) – (x2 + 7 – 8x)
(b)
The perimeter of a triangle can be represented by the expression 3x2 – 7x + 2.
Write a polynomial that represents the measure of the third side.
x2 – x – 4
2x2 – 10x + 6
2
MULTIPLYING WITH EXPONENTS (DAY 2)
POWERS OF THE SAME BASE
We know that:
m2 =
m3 =
So then…..
m2  m3 = (m  m) 
Similarly…..
c2  c4 = (c  c) 
(m
(c
 m  m) =
 c  c  c) =
Exponent Rule for Multiplying with the same Base:
xa

xb = x
You must have the same base exponents are _____________ together!
1.
x5  x4 = ___________
6.
c (c5) = ___________
2.
103  102 = ___________
7.
24  25  2 = ___________
3.
b6 (b) = ___________
8.
(q2)(2q4) = ___________
4.
m4a  m3a = ___________
9.
(9w2x8)(w6x4) = _____________
5.
z3  z4  z5 = ___________
10.
(14fg2h2)(-3f4g2h2) = ________________
11.
Write expressions for the areas of the two rectangles, separately, in the figures given
below:
3
FINDING THE POWER OF A POWER
Exponent Raised to an Exponent Rule:
(xa)c = x
The base remains the same, and the exponents are ________________________ together.
Option 1: Separate EACH piece of the BASE and raise it to the indicated Exponent, then
simplify.
(3x 2 y)3  (
)3 (
)3 (
)3 
Option 2: Write the expression the number of times indicated by Exponent #, then simplify.
(3x 2 y)3  (
)(
)(
)
Simplify the following:
1.
(a4)2 =
______________________
8.
(5  23)4 =
______________________
2.
(y6)3 =
______________________
9.
(22  32)4 =
______________________
3.
(2a4)2 =
______________________
10.
(xy3)2 =
______________________
4.
(x4)3 =
______________________
11.
(24  32)3 =
______________________
5.
(x3)4 =
______________________
12.
(3a2b4 )3 =
______________________
6.
(rs)3
______________________
13.
4 2 
 a  
5 
7.
(x3y4)2 =
______________________
14.
(2a3)4(a3)5 = ______________________
3
=
4
______________________
MULTIPLYING POLYNOMIALS (DAY 3)
Procedure for Multiplying Polynomials:
1. Multiply the coefficients
2. Multiply powers with same base by ADDING EXPONENTS
3. If expression has ( ) must distribute to simplify expression.
Simplify the following:
1.
(8xy)(3xz)
2.
(5x 2 y 3 )(2xy 2 )
4.
–5c2(15c – 4c2)
5.
10(2x –
7.
The length of a rectangle is represented by 5y – 7 and the width is represented by
3y. Determine the area of the rectangle.
1
x)
5
3.
–5(4m – 6n)
6.
5(d – 3 + d2) – 10d
What if you had to distribute a binomial to a binomial? Or a binomial to a trinomial?
Apply the same concept of multiplying, but do it more than once! Take turns!
Procedure:
Use the distributive property to simplify:
(x + 4)(x + 5)
This can be rewritten as:
x(x + 5) + 4(x + 5)
Let’s finish the problem:
5
[Show Arrows]
Simplify the following:
1)
(x + 2)(x + 3)
2)
(x – 4)(x – 2)
3)
(2x + 5)(3x – 4)
4)
(2z – 1)(3z2 + 1)
5)
z(2z + 1)(3z – 2)
6)
(x – 1)(x2 – x + 1)
7)
(b + 2)(b2 – 3b + 7)
8)
(y2 + 2y – 1)(y + 1)
9)
The length of a rectangle is represented by 5y2 – 7 and the width is represented by
3y3. Determine the area of the rectangle.
6
DISTRIBUTION (CONTINUED) – DAY 4
SPECIAL CASES:
Case 1: Binomials that have the same letters and variables in the same order, but
different middle signs are called _______________________. They are unique because
when you distribute them the middle terms cancel each other out.
1.
(x + 7)(x – 7)
2.
(4x + 3)(4x – 3)
4.
(3 + d)(3 – d)
Simplify the following:
3.
(10w – 1)(10w + 1)
Case 2: When squaring terms in parentheses you must square the ENTIRE term, which
means that you must WRITE IT_______________________, then distribute!!!
Simplify:
5.
(a + 1)2
6.
(5 + b)2
7.
(3x – 2)2
8.
(a + b)2
7
(s  3)2  s2  9
9.
Describe and correct the error shown here:
10.
(c2 + 2)(c + c2 – 3)
12.
The diagram shows a sandbox and the frame that surrounds it.
11.
(x + 2)(x2 + 3x + 5)
Write a polynomial that represents the area of the sandbox.
x+3
x+5
x+1
x-1
Write a polynomial that represents the area of the frame that surrounds the sandbox.
8
DIVIDING WITH EXPONENTS (DAY 5)
Using the Division Rule:
x3
x5
VS. Canceling Method,

Dividing Rule for Exponents:
x3
x5

xxx

xxxxx
Canceling Method for Exponents:
xa  xb = x
1. Determine where the base will
remain in fraction (top/bottom)
Base remains the same, and the exponents
2. Subtract/Reduce the small exponent
out of the larger exponent.
are _____________________ from each other.
Why would you choose the canceling method over the Division Rule?
Negative Exponents:
Basic Formula:
Procedure to change NEGATIVE Exponents into POSITIVE Exponents:
x–n = 1n
x
Simplify the following with positive exponents only.
1.
5.
x9
2.
x5
3 
3 3
35  34
6.
y10b
3.
y 3b
24a5
7.
 3a2
9
w6
w10
- 20x 3 y 5 z 2
 5x 7 y 3
4.
8.
a5  a2
a15
58
516  5
9.
13.
c 2a
c
10.
b
1
14.
25
8a  7 b 2
3
32c d
11. 3x 5
6
9a 8b 2
15.
27a10b
12. (2a)4
(x 6 )5
x 20  x 15
16.
4g2hc 5
6g  8h3 c
What would happen if we ended up with a Zero Exponent?
17.
x2
x
2
18.
40
20qr 2 t 5
21.
4q 0 r 4 t  2
10
19.
-4(x2)0
22.
16x y 
4x y z 
1 0
2
0
20.
4
2
30 x
DIVIDING POLYNOMIALS (DAY 6)
Recall: Simplify the following:
1.
24a5
 3a2
2.
- 20x 3 y 5 z 2
 5x 2 y 3
DIVIDING A POLYNOMIAL BY A MONOMIAL
STEPS:
Simplify:
1.
14x  7
7
2.
tr  r
r
3.
8c2  12d2
4
4.
p  prt
p
5.
28x 2  14
7
6.
7.
3a4 b2  183a2 b2
8.
 18a2 b2
11
 27y2  9y3
9y2
18x 3  12x 2  6x
 6x
9.
The volume of a rectangular prism (box) is represented by 18x 3  12x 2  6x . The
height of the box is 6ft and the width is x feet. Find the length of the box in terms of
x.
10.
The blueprint of a sandbox on a playground shows that it has an area of 45x 5  15x .
Using the picture below determine the width of the sandbox expressed as a
binomial in terms of x.
5x
11.
A soup can holds a volume represented by 18x 5  . The radius of the soup can is
represented by 3 x . Using the volume of a cylinder formula, V   r 2h , find the
height of the cylinder in terms of x.
12.
A swimming pool holds a volume represented by 6x 6  15x 5 . If the depth of the
pool is represented by 3 x 3 and the width is represented by x 2 , find the length of
the pool expressed as a binomial in terms of x.
12