5 .3 – polynomials and polynomial functions
DESCRIPTION
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 PresentationTRANSCRIPT
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.
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
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
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 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
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 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
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 9 123
x x xx
8
3
213xx
57x
6
3
93xx
4
3
123xx
33x 4x
5.4 – Multiplying PolynomialsSpecial Products
Extra Problems
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
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