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Algebra Chapter 8 Notes: Sargent Fall 2010

1

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


2

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.) (2p3 6p2 10p) (9p3 11p2 3p)

9.) (2x3 5x2 3x) (x3 8x2 11) 10.) (v3 6v2 v) (9v3 7v2 3v)

11.) (30d3 29d2 3d) (2d3 d2 )

Review Topics:

1.) Solve for x:

3

2 5

x x

2.) Find f(-2) if f(x) = x2 – 3


3

3.) Find the slope and y-intercept of the line:

4x – y = -6

4.) Simplify: (3x2y

4)(-2xy

3)

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 10

x

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

y


4

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.) 3t2 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.) 5d3 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


5

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


6

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)


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 2p)2

12.) (9c 8)2 13.) (4k 3)2 14.) (2y 11)2

Notes #10

8-8: Factoring 4-termed Polynomials

Steps: 12x3 + 15x – 4x

2 – 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.) 12p4 10p3 36p2 30p


8

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


9

Notes #11:

8-5: Factoring Trinomials using the X-Box Method

Steps:

Write in descending order

Take out a GCF

Draw an X and a Box.

Fill them in like this:

Factor and Check like before

Factor and check using FOIL.

1.) 2 7 12x x

2.) 2 13 36x x

3.) 2 9x

4.) 2 9 20x x

5.) 22 32x

6.) 2 8 15x x


10

7.) 2 25x

8.) 23 21 36x x

9.) 2 4 12x x

10.) 25 25x

11.) 2 11 28x x

12.) 4 100p

13.) 3 23 3x x x

14.) 2 28 15m mn n

15.) 2 23 15 18a ab b


11

Simplify: (_____________ first!!)

22.) 2

2

7 7

3 4

x x

x x

23.)

2

2

4

3 10

x

x x

24.) 2 3

2 2

6 10

5 10 4 3

x x x

x x x x

25.)

2 4

2 2

3 15 6

6 5 1

m m m

m m m

16.) 2 24 16a b

17.)

2 25 14a ab b

18.) 2 230x xy y

19.) 3 22 4 8x x x 20.)

2 249g h

21.) 2 26 8p pq q


12

Notes #12

8-6: Factoring Trinomials of the Type ax2 + bx + c

Steps:

Write in descending order

Take out a GCF

Draw an X and a Box.

Fill them in like this:

Factor and Check with FOIL

Factor.

1.) 23 5 2x x

2.) 24 4 15n n

3.) 28 10 3x x

4.) 26 41 7x x


13

5.) 22 4 6x x

6.) 27 20 3m m

7.) 25 13 6y y

8.) 29 15 6a a

Simplify:

9.) 2

2

2 5 3

9

m m

m

10.)

2

3 2

2 6 9

6 2 3

m m m

m m m


14

Notes #13: Factoring Review

Look for a GCF

Factor using the Box (4-terms) or X-Box (2 or 3 terms)

Check using FOIL

If the answer includes the product of two identical binomials, write that part as (______)2

Factor Completely. Check with FOIL.

1.) 7x2 – 7 2.) x

4 + 4x

3 – 45x

2 3.) x

3 + 2x

2 – 4x – 8

4.) x2 8x 16 5.) n2 16n 64 6.) 23 18 27m m

7.) 3 26 6y y y 8.) 3x

2 + 14x – 5 9.) 3x

2 + 6x – 105

10.) 6x2 – 54 11.) 9g

2 12g 4 12.) 23 6 3a a


15

13.) 2x2 + 7x + 3 14.) 23 2 8n n 15.) 10x2 40

16.) 4w2 49 17.) 4x2 121 18.) 3c2 75

Simplify:

19.) (-z3 + 3z) + (-z

2 – 4z – 6) 20.) (5x

2 + 7x – 4) – (4x

2 – 2x)

21.) (2c – 3)2 22.) -5x(3x – 2)(x + 1)

23.) 4( 3)

10( 3)

x

x

24.)

6( 1)( 5)

15(1 )( 3)

x x

x x

25.) 6 18

5 15

x

x

26.)

2

2

4

2

x

x x


16

Notes #14: Review Simplifying Rational Expressions:

Factor all expressions (numerators and denominators) ___________________

[Make sure you have factored correctly by _________________!!!

If a division problem, _________ the second fraction and write as a

___________________ problem.

Reduce by canceling out ___________ ____________ (terms in parentheses) from

the numerator and denominator

Look to cancel opposites; leave a -1 behind. Ex: 4

4

x

x

=

Re-write answer as a new and reduced fraction

Simplify:

1.)

2

2

2 3

1

x x

x

2.)

2

2

5 6

6 8

m m

m m

3.)

2

4 2

2

5 4

w w

w w

4.)

3 2

2

4 4

3 2

y y y

y y

5.)

2

3

32 8

5 20

k

k k

6.)

2 9

3

h

h


17

7.) 6 12 3 9

5 15 2 4

x x

x x

8.)

3 12 5 20

2 20 2 4

x x

x x

9.)

2 2

2

4 1

2 3 2

x x

x x x

10.)

2

2

2 8 2

2 4 10

x x

x x x

11.) 2 3

5 2 6

9 15

y y

y y

12.)

3

2 2

6 12

16 5 4

y y

y y y


18

13.)

2 2 22 5 3

2 1

c c c d

c d c

14.)

2 2

3 2

4 1 2 5 3

6 6 3 9

c c c

c c c

15.)

3 2 2

2 2

2 4 8 1

3 2 2 3

y y y y

y y y y


19

Algebra 1: Chapter 8 Study Guide Name: __________________

Per: ______ Date: ______

For #1-3, find the degree of each polynomial:

1.) -4x2y

3 + 7xy

2.) 7

3.) 6a2 – 3a + 1

For #4-6, write each polynomial in standard form. Then name the polynomial based on its

degrees and number of terms.

4.) 4m – 8 + 9m3

5.) 7 – 5x 6.) 8p2 – 3 + 4p

2 – p

For #7-10, simplify. Write each answer in standard form

7.) (5x3 – 6x + 7) + (4x

2 – 7x + 3)

8.) (3x2 – 9xy + 13) + (x

2 + 4xy – 2)

9.) (4a4 – 7a

3 – 9a) – (11a

2 – 8a

2 + 7a)

10.) 4(2x – 3y) – 5(2y – 3x)

Multiply. Leave your answer in standard form.

11.) 3x(5x2 + 2x – 11)

12.) -8x2y

3(2x

3– 5xy) 13.) (2x – 1)

2

14.) (x + 3)(x2 + x + 2)

15.) (3a + 2b)2 16.) 4w(w – 3)(w + 5)

Factor each polynomial completely. Show your check where indicated.

17.) 14x2y

5 – 21xy

3

18.) 5x2y

5 + 10x

3y

4 + 25xy

19.) x2 – 25


20


20.) m3 – 3m

2 – 5m + 15

21.) x3 + 3x

2 – 4x – 12

22.) x2 + 9x + 14

23.) x2 – 7x – 18 24.) 3x

2 – 6x + 3

25.) d2 + 13de + 22e

2

(check):

26.) 6x2 – 7x – 5

(check):

27.) 2y2 – 13y + 6

(check):

28.) 5x 2 + 40x + 80 29.) x

2 – 5x + 7

30.) 3x2 – 75


21


31.) y2 + 49 32.) 8m

2 + 14m + 6

Simplify.

33.) 2

2

3 2x x

x x

34.) 2

3 2

4

2 4

x

x x

35.) 3 2

2

2 9 18

6 16

m m m

m m

36.) 25 5 3

9 9 10 20

x x

x x

37.) 2

2 2

5 24 3

6 18 9 8

x x x

x x x x

38.) 2 2

3 2

3 10 25

12 8

y y y

y y

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