5.3 – Polynomials and Polynomial FunctionsDefinitions

Coefficient: the numerical factor of each term.

Constant: the term without a variable.

Term: a number or a product of a number and variables raised to a power.

Polynomial: a finite sum of terms of the form axn, where a is areal number and n is a whole number.

2 23, 5 , 2 , 9x x x y

2 29, ,5 2x x x y

3, 6, 5, 32

3 215 2 5x x 6 5 321 7 2 6y y y y

Definitions

Monomial: a polynomial with exactly one term.

Binomial: a polynomial with exactly two terms.

Trinomial: a polynomial with exactly three terms.

8,x

2 ,ax

2 8,x x

9 ,m 29x y42 ,x,rt

3,r 25 2 ,x x 22 9x x y

5 3 3,r r 25 2 7x x

5.3 – Polynomials and Polynomial Functions

Definitions

25x

The Degree of a Term with one variable is the exponent on the variable.

The Degree of a Term with more than one variable is the sum of the exponents on the variables.

27x y The Degree of a Polynomial is the greatest degree of the terms of the polynomial variables.

32 3 7x x

2, 42x 14, 9m

3, 4 22x y 106, 5 49mn z

4 2 2 32 5 6x y x y x 63,

5.3 – Polynomials and Polynomial Functions

Practice Problems

3 25 4 5x x x

Identify the degrees of each term and the degree of the polynomial.

2 4 5 33 2 9 4a b ab b

5 4 4 5 3 34 5 6 2x y x y x y xy

3 2 1

3

9 9 6 2

9

6 6 3 0

6

5.3 – Polynomials and Polynomial Functions

Practice Problems

Evaluate each polynomial function

23 11 0 3 1 10 3 10 7

26 11 23 3 0

6 9 33 20

54 33 20 87 20 67

5.3 – Polynomials and Polynomial Functions

103 2 xxf 1f

20116 2 yyyg 3g

5.4 – Polynomials and Polynomial FunctionsMultiplication

Multiplying Monomials by Monomials

Examples:

10 9x x

3 78 11x x

45x x

290x

1088x

55x

Multiplying Monomials by Polynomials

Examples:

3 25 3 2x x x

24 4 3x x x

48 7 1x x

34x 216x 12x

556x 8x

515x 45x 310x

5.4 – Polynomials and Polynomial FunctionsMultiplication

Multiplying Two Polynomials

Examples:

25 10 3x x x

24 5 3 4x x x

3x 210x 3x 25x 50x 153x 215x 47x 15

312x 216x 23x 4x 15x 20 312x 213x 11x 20

5.4 – Polynomials and Polynomial FunctionsMultiplication

5.4 – Multiplying PolynomialsSpecial Products

Multiplying Two Binomials using FOIL

7 4x x

3 4x x

First terms Last termsInner termsOuter terms

2x 4x 7x 28

2x 7x 12

2x 3x 28

2x 4x 3x 12

Multiplying Two Binomials using FOIL

22 3 4y y

First terms Last termsInner termsOuter terms

22 4y

32y 28y 3y 12

32y 28y 3y 12

2 24 4y y

4y 24y 16 24y 4y 28y 16

5.4 – Multiplying PolynomialsSpecial Products

Squaring Binomials

24 5x 216x

216 40 25x x

2 2 2

2 2 2

2

2

a b a ab b

a b a ab b

23 6x 29x

29 36 36x x

2 20x 25

2 18x 36

5.4 – Multiplying PolynomialsSpecial Products

Multiplying the Sum and Difference of Two Binomials

2 2a b a b a b

9 9x x 2x 2 81x

3 5 3 5x x 29x 29 25x

9x 9x 81

15x 15x 25

5.4 – Multiplying PolynomialsSpecial Products

88 xx 642 x

135135 xx 16925 2 x

Dividing by a Monomial

where 0a b a b

cc c c

8 6 4

3

21 9 12

3

x x x

x

8

3

21

3

x

x

57x

6

3

9

3

x

x

4

3

12

3

x

x

33x 4x

5.4 – Multiplying PolynomialsSpecial Products

285

5

Review of Long Division

783 4

4 7835 2855

2535

7

350

438362

19

3

5

203

6.4 – Dividing PolynomialsLong Division

4

3195

4

3195

35125 2 xxx

x

xx 52 x7 35

7

357 x

0

5x 7x 35122 xx

6.4 – Dividing PolynomialsLong Division

5

35122

x

xx

72812 2 xxx

x4

xx 48 2 x6 7

3

36 x

4

12 x

12

434

xx 728 2 xx

12

4

x

6.4 – Dividing PolynomialsLong Division

12

728 2

x

xx

15023 2 xxx

x2

xx 62 2 x6 15

6

186 x

3

3x

3

362x

x 152 2 x

3

3

x

6.4 – Dividing PolynomialsLong Division

3

152 2

x

x

5.7 - Factoring Polynomials

Factoring Trinomials

4 5x x

2x bx c 2 5 4 20x x x 2 9 20x x

4 5x x 2 209x x

20 is the result of the product of 4 and 5.

9 is the result of the sum of 4 and 5.

Factors of 20 are: 1, 20 2,10 4, 5

Factoring Trinomials 2x bx c 2 10 24x x

Factors of 24 are: 1, 24 2, 12 3, 8 4, 6

x x

4 6x x

5.7 - Factoring Polynomials

Factoring Trinomials 2x bx c 2 27 50x x

Factors of 50 are: 1, 50 2, 25 5,10

x x

2 25x x

5.7 - Factoring Polynomials

Factoring Trinomials 2x bx c

Factors of 9 are: 1, 9 3, 3

4 x x

4 3 3x x

24 24 36x x

24 6 9x x

5.7 - Factoring Polynomials

Factoring Trinomials 2ax bx c

Factors of 12 are: 1, 12 2,6 3, 4

x x

4 2 3x x

22 11 12x x

Factors of 2 are: 1, 2

5.7 - Factoring Polynomials

Factoring Perfect Square Trinomials

6 6x x

2 12 36x x

Not a perfect square

36132 xx

94 xx

26x

x x x x

5.7 - Factoring Polynomials

Factoring Perfect Square Trinomials 2 18 81x x

9 9x x

29 42 49x x

3 7 3 7x x

29x 273 x

x x x x

5.7 - Factoring Polynomials

Factoring the Difference of Two Squares

3 1 3 1s s 29 1s

2 81p

24 100x

9 9p p

Not the difference

5.7 - Factoring Polynomials

Factoring the Difference of Two Squares

3 38 8

5 5c c

2 9

6425

c

2 2121 49x y 11 7 11 7x y x y

5.7 - Factoring Polynomials

Factoring the Sum or Difference of Two Cubes5.7 - Factoring Polynomials

2233

2233

babababa

babababa

273 xx 3 2x x3 9

Factoring the Sum or Difference of Two Cubes5.7 - Factoring Polynomials

2233

2233

babababa

babababa

83 yy 2 2y y2 4

Factoring the Sum or Difference of Two Cubes5.7 - Factoring Polynomials

2233

2233

babababa

babababa

33 64yx

x 4 2x xy4 16 y 2y

Factoring the Sum or Difference of Two Cubes5.7 - Factoring Polynomials

2233

2233

babababa

babababa

23227 aba

3 9 b3 b2b

2a 327 b

2a

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Presentation

Multiplying Two Polynomials

Examples:

105 xx 2x x10 x5 50

2x x15 50

4354 xx 212x x16 x15 20

212x x 20

5.4 – Polynomials and Polynomial FunctionsMultiplication

Multiplying Two PolynomialsExamples:

23 2 5 4x x x 32x 25x 4x 26x 15x 12

32x 2x 11x 12

5.4 – Polynomials and Polynomial FunctionsMultiplication

Multiplying Two PolynomialsExamples:

23 2x y

3 2 3 2x y x y 2x 3xy 2x 3xy 6y29y 2x 6y 4

2x 6xy 4x 29y 12y 4

5.4 – Polynomials and Polynomial FunctionsMultiplication