5 .3 – Polynomials and Polynomial Functions

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5.3 – Polynomials and Polynomial Functions Definitions Coefficient : the numerical factor of each term. stant : the term without a variable. Term : a number or a product of a number and variables raised to a power. mial : a finite sum of terms of the form ax n , where a umber and n is a whole number. 2 2 3, 5 , 2,9 x x xy 2 2 9 , , 5 2 x x xy 3, 6, 5, 32 3 2 15 2 5 x x 6 5 3 21 7 2 6 y y y y

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5 .3 – Polynomials and Polynomial Functions. Definitions. Term : a number or a product of a number and variables raised to a power. Coefficient : the numerical factor of each term. Constant : the term without a variable. Polynomial : a finite sum of terms of the form a x n , where a is a - PowerPoint PPT Presentation

Transcript of 5 .3 – Polynomials and Polynomial Functions

Page 1: 5 .3 – Polynomials and Polynomial Functions

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

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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

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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

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Practice Problems

3 25 4 5x x x

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

3 2 1

3

9 6 2

6 3 0

5.3 – Polynomials and Polynomial Functions

4𝑎2𝑏4+3𝑎3𝑏5−9𝑏4+48

8

4 𝑥5 𝑦 4+5 𝑥4 𝑦6−6 𝑥3 𝑦 3+2𝑥𝑦10

10

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Practice ProblemsEvaluate 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

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5.4 – Polynomials and Polynomial FunctionsMultiplication

Multiplying Monomials by Monomials

Examples:

10 9x x

3 78 11x x

45x x

290x

1088x

55x

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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

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Multiplying Two PolynomialsExamples:

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

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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

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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

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Squaring Binomials

24 5x 216x216 40 25x x

2 2 2

2 2 2

2

2

a b a ab b

a b a ab b

23 6x 29x29 36 36x x

2 20x 25

2 18x 36

5.4 – Multiplying PolynomialsSpecial Products

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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

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Dividing by a Monomial

where 0a b a b cc c c

8 6 4

3

21 9 123

x x xx

8

3

213xx

57x

6

3

93xx

4

3

123xx

33x 4x

5.4 – Multiplying PolynomialsSpecial Products

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Extra Problems

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Multiplying Two PolynomialsExamples:

105 xx 2x x10 x5 50

2x x15 50

4354 xx 212x x16 x15 20

212x x 20

5.4 – Polynomials and Polynomial FunctionsMultiplication

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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

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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