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Name: Date:__________ Period: __________
CHAPTER 8: POLYNOMIALS AND FACTORING
Notes #6
8-1: Adding and Subtracting Polynomials
A. Describing polynomials
A ____________________ is an expression that is a number, a variable, or a product of a
number and one or more variables.
Ex:
The _____________ of a monomial is the sum of the exponents of its variables. For a
nonzero constant, the degree is ___. Zero has ____ degree.
Find the degree of each monomial.
1.) 2
3x 2.) 7x2y3 3.) 4
A ____________________ is a monomial or a sum and/or difference of monomials.
Standard form of a polynomial means that the degrees of its monomial terms
______________ from left to right.
The ____________ of a polynomial in one variable is the same as the degree of the
monomial with the greatest exponent.
Ex: 3x4 5x2 7x 1
Polynomial Degree Name Using
Degree Number of
Terms
Name Using Number of
Terms
7x 4
3x2 2x 1
4x3
9x4 11x
5
Write each polynomial in standard form. Then name each polynomial based on its degree and
number of its terms.
4.) 5 2x 5.) 3x4 4 2x2 5x4 6.) 3y 4 y3
B. Adding and subtracting polynomials
You can add polynomials by adding or subtracting like terms. You can add or
subtract vertically or horizontally.
Simplify.
7.) (4x2 6x 7) (2x2 9x 1) 8.) (2 p3 6 p2 10 p) (9 p3 11p2 3p)
9.) (2x3 5x2 3x) (x3 8x2 11) 10.) (v3 6v2 v) (9v3 7v2 3v) 11.) (30d 3 29d2 3d) (2d 3 d2 ) Review Topics: 1.) Solve for x:
3
2 5
x x
2.) Find f(-2) if f(x) = x2 – 3
3.) Find the slope and y-intercept of the line:
4x – y = -6
4.) Simplify: (3x2y4)(-2xy3)2
5.) Solve: 5 1 14x 6.) Solve for a and b:
9
3 2 17
a b
a b
7.) Find the x-intercept and the y-intercept of the line:
3x + 4y = -6
8.) Graph the line: 2x – 5y = 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x
-10
-9-8
-7
-6-5
-4
-3
-2
-1
12
3
45
6
7
89
10y
Notes #7/8 8-2: Multiplying and Factoring
A. Distributing a monomial
Simplify each product.
1.) 4y2 (5y4 3y2 2) 2.) 2x(x2 6x 5) 3.) 7h(3h2 8h 1)
B. Factoring a monomial from a polynomial
Find the common factor of the terms of each polynomial.
4.) 4x3 12x2 8x 5.) 5v5 10v3 6.) 3t 2 18
To factor a polynomial completely, you must factor until there are no common factors
other than ____.
Factor. Distribute to check your answer.
7.) 3x3 12x2 15x 8.) 8x2 12x
9.) 6m3 12m2 24m 10.) 5d 3 10d
11.) 5 4 236 48 24g g g 12.) 214 21y xy
13.) 3 2 2 232 24 16x y x y xy 14.) 3 5 2 481 18a b a b
C. Applications to Reducing Fractions
Factor the ______________________ and _________________________ completely
Cancel common terms in the __________________________ and common factors ( )
Ex: 3 4
7
16
18
w z
wz
Ex:
5 5
12 12
x
x
Simplify. (Factor first!!)
15.) 5 3
8 2
50
30
d e
d e 16.)
3 8
8 12
64
40
x y
x y
17.)
5 0
5
49
14
t u
t
18.) 27 9
9 3
a
a
19.) 7 14
8 8
m
m
20.) 2
3 2
24 30
48 60
b b
b b
21.) 2 212 24
20 40
a b a
ab a
22.) 2
2
12 8
18 12
x x
x y xy
Notes #9
8-3: Multiplying Binomials
A. Multiplying two binomials
One way to organize multiplying two
binomials is to use FOIL, which stands for:
o F
o O
o I
o L
Ex: (4x – 1)(x + 3)
Another way to multiply polynomials is to use
boxes. Multiply each monomial together and
put their product in their shared box. Combine
like terms and write your answer as a
polynomial in descending order.
Ex: (4x – 1)(x + 3)
Simplify.
1.) (2x 3)(x 4) 2.) (5 2)(8 1)m m
3.) (9a 8)(7a 4) 4.) (6 7)(6 7)h h
5.) (3x 4)(2x 5) 6.) (3 4)(3 4)x x
B. Multiplying a trinomial and a binomial
Simplify the product.
7.) (4x2 x 6)(2x 3) 8.) (6n 8)(2n2 n 7)
8-4: Multiplying Special Cases
A. Finding the square of a binomial (___________ in disguise!)
The square of a binomial:
o (a b)2
o (a b)2
Find each square.
9.) (t 6)2 10.) (x 7)2 11.) (7m 2 p)2
12.) (9c 8)2 13.) (4k 3)2 14.) (2y 11)2
Notes #10
8-8: Factoring 4-termed Polynomials
Steps: 12x3 + 15x – 4x2 – 5
Write in standard form (descending order)
Check for a GCF
Draw a 2x2 box and fill it in with the four terms
Find the GCF for each row and each column. Take the sign
(positive/negative) of the leading term
Write your answer as (______)(______)
FOIL to check
Factor using the Box Method. FOIL to check..
1.) 4n3 8n2 5n 10 2.) 5t 4 20t 3 6t 24
3.) 2w3 w2 14w 7 4.) 12 p4 10 p3 36 p2 30 p
5.) 45m4 9m3 30m2 6m 6.) 3 25 20 4x x x
7.) 4 312 3 12 3x x x 8.)
4 3 25 5 30 30m m m m
Simplify: ( _________________ first!!)
9.) 2 3
2
24
24 8
x y
x y xy 10.)
3 2
3
4 12 3
4 12
m m m
m m
11.) 7 2
11 3 12.)
5 ( 1)
3 6
x x
13.)
2
2
5 63 4
25( 1) 6 4 6
xx
x x x
14.) 3
2
12 8
7 14
xy x
mn m 15.)
6 5 10
1 2 4
w w
w w
16.) 3 12 9 9
5 2 2
n n
n n
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