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 ) 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 #7 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

Simplify. (Factor first!!)

15.) 3 7

6 3

36

24

d e

d e 16.)

2 8

8 11

68

40

x y

x y

17.)

5 0

5

49

21

x y

x

18.) 27 9

18 6

a

a

19.) 5 15

4 12

m

m

20.) 2

3 2

24 30

48 60

b b

b b

21.) 2 215 30

3 6

a b a

ab a

22.) 2

2

12 8

18 12

x x

x y xy

Review Topics: 1.) 3 1 2 5x x 2.) 3 1 2 5x x

3.) 22 1 3 2x x x 4.) 22 1 3 2x x x

5.) 6 5 3x x 6.) 6 5 3x x

Notes #8 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

Notes #9:

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

Steps:

Write in descending order

Take out a GCF

If the leading term is negative,

factor out a negative sign

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

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

Notes #10: More X Box Method

Steps:

Write in descending order

Take out a GCF

If the leading term is negative,

factor out a negative sign

Draw an X and a Box.

Fill them in like this:

Factor and Check like before

1.) 2 24 16a b

2.) 2 25 14a ab b

3.) 2 230x xy y

4.) 3 22 4 8x x x 5.) 2 249g h

6.) 2 26 8p pq q

Simplify: (_____________ first!!)

7.) 2

2

7 7

3 4

x x

x x

8.) 2

2

4

3 10

x

x x

9.) 2

2

2 3

1

x x

x

10.)

2

2

5 6

6 8

m m

m m

11.) 2

4 2

2

5 4

w w

w w

12.)

3 2

2

4 4

3 2

y y y

y y

Notes #11: Factoring Review Factor out a GCF or leading negative sign

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.) x4 + 4x3 – 45x2 3.) x3 + 2x2 – 4x – 8

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

7.) 3 26 6y y y 8.) 3x2 + 14x – 5 9.) 3x2 + 6x – 105

10.) 6x2 – 54 11.) 9g2 12g 4 12.) 23 6 3a a

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) + (-z2 – 4z – 6) 20.) (5x2 + 7x – 4) – (4x2 – 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

Notes 12: Chapter 8 & 11. 1 Review For #1-2, write each polynomial in standard form. Then name the polynomial based on its degrees and number of terms. 1.) 8x2 + 7x3 – 9

2.) 10x + 3 3.) 10p2 – 3 + 5p

For #3-7, simplify. Write each answer in standard form 3.) (11x3 – 5x + 2) – (6x2 – 2x + 5)

4.) (3h – 8)2

5.) (x – 3)(x2 + 2x – 1) 6.) 3(2y – 5x) – 5(2y – 3x)

7.) 4m(m – 3)(m + 2)

For #8-13, factor each polynomial completely. If it cannot factor, write prime. Show your check where indicated. 8.) 15a2b7 – 10ab3 9.) x2 + 25 10.) 8d2 – 50

11.) x3 + 5x2 – 4x – 20

12.) 6y2 – 7y – 5 (check):

13.) 3x2 – 10xy – 8y2 (check):

For #14, simplify.

14.) 3 2

2

6 6

2 2

x x

x