6.3 – Evaluating Polynomials. degree (of a monomial) 5x 2 y 3 degree =

6.3 – Evaluating Polynomials

degree (of a monomial) 5x 2 y 3 degree =

degree (of a monomial) 5x 2 y 3 degree = 5

degree (of a monomial) 5x 2 y 3 degree = 5 degree

degree (of a monomial) 5x 2 y 3 degree = 5 degree (of a polynomial w/1 variable)

degree (of a monomial) 5x 2 y 3 degree = 5 degree (of a polynomial w/1 variable) - highest exponent of a single term

degree (of a monomial) 5x 2 y 3 degree = 5 degree (of a polynomial w/1 variable) - highest exponent of a single term Ex. 1 State the degree and leading coefficient of the polynomial. 7x 4 + 5x 2 + x – 9

degree (of a monomial) 5x 2 y 3 degree = 5 degree (of a polynomial w/1 variable) - highest exponent of a single term Ex. 1 State the degree and leading coefficient of the polynomial. 7x 4 + 5x 2 + x – 9 degree =

degree (of a monomial) 5x 2 y 3 degree = 5 degree (of a polynomial w/1 variable) - highest exponent of a single term Ex. 1 State the degree and leading coefficient of the polynomial. 7x 4 + 5x 2 + x 1 – 9 degree =

degree (of a monomial) 5x 2 y 3 degree = 5 degree (of a polynomial w/1 variable) - highest exponent of a single term Ex. 1 State the degree and leading coefficient of the polynomial. 7x 4 + 5x 2 + x 1 – 9x 0 degree =

degree (of a monomial) 5x 2 y 3 degree = 5 degree (of a polynomial w/1 variable) - highest exponent of a single term Ex. 1 State the degree and leading coefficient of the polynomial. 7x 4 + 5x 2 + x – 9 degree = 4

degree (of a monomial) 5x 2 y 3 degree = 5 degree (of a polynomial w/1 variable) - highest exponent of a single term Ex. 1 State the degree and leading coefficient of the polynomial. 7x 4 + 5x 2 + x – 9 degree = 4 lead coeff. =

degree (of a monomial) 5x 2 y 3 degree = 5 degree (of a polynomial w/1 variable) - highest exponent of a single term Ex. 1 State the degree and leading coefficient of the polynomial. 7x 4 + 5x 2 + x – 9 degree = 4 lead coeff. = 7

Funtional Values

Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5)

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) =

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) = (-5) 3

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) = (-5) 3 + 4(-5) 2

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) = (-5) 3 + 4(-5) 2 – 5(-5)

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) = (-5) 3 + 4(-5) 2 – 5(-5) = -125

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) = (-5) 3 + 4(-5) 2 – 5(-5) = -125 + 4(25)

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) = (-5) 3 + 4(-5) 2 – 5(-5) = -125 + 4(25) – 5(-5)

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) = (-5) 3 + 4(-5) 2 – 5(-5) = -125 + 4(25) – 5(-5) = -125 + 100

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) = (-5) 3 + 4(-5) 2 – 5(-5) = -125 + 4(25) – 5(-5) = -125 + 100 + 25

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) = (-5) 3 + 4(-5) 2 – 5(-5) = -125 + 4(25) – 5(-5) = -125 + 100 + 25 = -25 + 25

Funtional Values Ex. 2 Find the following functional values if p(x) = x 3 + 4x 2 – 5x a. p(-5) p(-5) = (-5) 3 + 4(-5) 2 – 5(-5) = -125 + 4(25) – 5(-5) = -125 + 100 + 25 = -25 + 25 = 0

b. p(3a)

p(x) = x 3 + 4x 2 – 5x

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) =

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a)

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a)

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 5(3a)

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a)

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a) p(x) = x 3 + 4x 2 – 5x

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a) p(x) = x 3 + 4x 2 – 5x 3[p(a)]

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a) p(x) = x 3 + 4x 2 – 5x 3[p(a)] 3[p(a)] =

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a) p(x) = x 3 + 4x 2 – 5x 3[p(a)] 3[p(a)] = 3

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a) p(x) = x 3 + 4x 2 – 5x 3[p(a)] 3[p(a)] = 3[

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a) p(x) = x 3 + 4x 2 – 5x 3[p(a)] 3[p(a)] = 3[a 3 + 4a 2 – 5a]

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a) p(x) = x 3 + 4x 2 – 5x 3[p(a)] 3[p(a)] = 3[a 3 + 4a 2 – 5a] = 3(a 3 )

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a) p(x) = x 3 + 4x 2 – 5x 3[p(a)] 3[p(a)] = 3[a 3 + 4a 2 – 5a] = 3(a 3 ) + 3(4a 2 )

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a) p(x) = x 3 + 4x 2 – 5x 3[p(a)] 3[p(a)] = 3[a 3 + 4a 2 – 5a] = 3(a 3 ) + 3(4a 2 ) – 3(5a)

b. p(3a) p(x) = x 3 + 4x 2 – 5x p(3a) = (3a) 3 + 4(3a) 2 – 5(3a) = (3) 3 (a) 3 + 4(3) 2 (a) 2 – 5(3a) = 27a 3 + 4(9)a 2 – 15a = 27a 3 + 36a 2 – 15a c. 3p(a) p(x) = x 3 + 4x 2 – 5x 3[p(a)] 3[p(a)] = 3[a 3 + 4a 2 – 5a] = 3(a 3 ) + 3(4a 2 ) – 3(5a) = 3a 3 + 12a 2 – 15a

Ex. 3 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x

Ex. 3 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x 1. As x +∞

Ex. 3 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x 1. As x +∞, f(x)

Ex. 3 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x 1. As x +∞, f(x) –∞

Ex. 3 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x 1. As x +∞, f(x) –∞ As x –∞

Ex. 3 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x 1. As x +∞, f(x) –∞ As x –∞, f(x)

Ex. 3 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x 1. As x +∞, f(x) –∞ As x –∞, f(x) –∞

Ex. 3 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x 1. As x +∞, f(x) –∞ As x –∞, f(x) –∞ 2. Even degree

Ex. 3 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x 1. As x +∞, f(x) –∞ As x –∞, f(x) –∞ 2. Even degree 3. 2 real zeros

Ex. 4 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x 1. As x +∞, f(x) As x –∞, f(x) 2. degree 3. real zero(s)

Ex. 4 For each graph: 1.Describe the end behavior 2.Determine if the degree is even or odd. 3.State the number of real zeros. y (solutions) a. x 1. As x +∞, f(x) +∞ As x –∞, f(x) –∞ 2. Odd degree 3. 1 real zero

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6.3 – Evaluating Polynomials. degree (of a monomial) 5x 2 y 3 degree =

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