6.3 – Evaluating Polynomials

degree (of a monomial)5x2y3 degree =

degree (of a monomial)5x2y3 degree =

degree (of a monomial)5x2y3 degree = 5

degree (of a monomial)5x2y3 degree = 5

degree

degree (of a monomial)5x2y3 degree = 5

degree (of a polynomial w/1 variable)

degree (of a monomial)5x2y3 degree = 5

degree (of a polynomial w/1 variable)- highest exponent of a single term

degree (of a monomial)5x2y3 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. 7x4 + 5x2 + x – 9

degree (of a monomial)5x2y3 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. 7x4 + 5x2 + x – 9

degree =

degree (of a monomial)5x2y3 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. 7x4 + 5x2 + x – 9

degree =

degree (of a monomial)5x2y3 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. 7x4 + 5x2 + x – 9

degree =

degree (of a monomial)5x2y3 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. 7x4 + 5x2 + x1 – 9

degree =

degree (of a monomial)5x2y3 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. 7x4 + 5x2 + x1 – 9x0

degree =

degree (of a monomial)5x2y3 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. 7x4 + 5x2 + x – 9

degree = 4

degree (of a monomial)5x2y3 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. 7x4 + 5x2 + x – 9

degree = 4lead coeff. =

degree (of a monomial)5x2y3 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. 7x4 + 5x2 + x – 9

degree = 4lead coeff. =

degree (of a monomial)5x2y3 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. 7x4 + 5x2 + x – 9

degree = 4lead coeff. = 7

Funtional Values

Funtional Values

Ex. 2 Find the following functional values if p(x) = x3 + 4x2 – 5x

Funtional Values

Ex. 2 Find the following functional values if p(x) = x3 + 4x2 – 5x

a. p(-5)

Funtional Values

Ex. 2 Find the following functional values if p(x) = x3 + 4x2 – 5x

a. p(-5)

Funtional Values

Ex. 2 Find the following functional values if p(x) = x3 + 4x2 – 5x

a. p(-5)

p(-5) =

Funtional Values

Ex. 2 Find the following functional values if p(x) = x3 + 4x2 – 5x

a. p(-5)

p(-5) =

Funtional Values

Ex. 2 Find the following functional values if p(x) = x3 + 4x2 – 5x

a. p(-5)

p(-5) = (-5)3

Funtional Values

Ex. 2 Find the following functional values if p(x) = x3 + 4x2 – 5x

a. p(-5)

p(-5) = (-5)3 + 4(-5)2

Funtional Values

Ex. 2 Find the following functional values if p(x) = x3 + 4x2 – 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) = x3 + 4x2 – 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) = x3 + 4x2 – 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) = x3 + 4x2 – 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) = x3 + 4x2 – 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) = x3 + 4x2 – 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) = x3 + 4x2 – 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) = x3 + 4x2 – 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) = x3 + 4x2 – 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)

b. p(3a)

p(x) = x3 + 4x2 – 5x

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) =

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) =

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) =

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2

b. p(3a)

p(x) = x3 + 4x2 – 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) = x3 + 4x2 – 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) = x3 + 4x2 – 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) = x3 + 4x2 – 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) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 5(3a)

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

c. 3p(a)

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

c. 3p(a)

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

c. 3p(a)

p(x) = x3 + 4x2 – 5x

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

c. 3p(a)

p(x) = x3 + 4x2 – 5x

3[p(a)]

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

c. 3p(a)

p(x) = x3 + 4x2 – 5x

3[p(a)]

3[p(a)] =

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

c. 3p(a)

p(x) = x3 + 4x2 – 5x

3[p(a)]

3[p(a)] = 3

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

c. 3p(a)

p(x) = x3 + 4x2 – 5x

3[p(a)]

3[p(a)] = 3[

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

c. 3p(a)

p(x) = x3 + 4x2 – 5x

3[p(a)]

3[p(a)] = 3[

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

c. 3p(a)

p(x) = x3 + 4x2 – 5x

3[p(a)]

3[p(a)] = 3[

b. p(3a)

p(x) = x3 + 4x2 – 5x

p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a)

= 27a3 + 4(9)a2 – 15a

= 27a3 + 36a2 – 15a

c. 3p(a)

p(x) = x3 + 4x2 – 5x

3[p(a)]

3[p(a)] = 3[

b. p(3a) p(x) = x3 + 4x2 – 5x p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a) = 27a3 + 4(9)a2 – 15a = 27a3 + 36a2 – 15a

c. 3p(a) p(x) = x3 + 4x2 – 5x 3[p(a)] 3[p(a)] = 3[a3 + 4a2 – 5a]

b. p(3a) p(x) = x3 + 4x2 – 5x p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a) = 27a3 + 4(9)a2 – 15a = 27a3 + 36a2 – 15a

c. 3p(a) p(x) = x3 + 4x2 – 5x 3[p(a)] 3[p(a)] = 3[a3 + 4a2 – 5a]

b. p(3a) p(x) = x3 + 4x2 – 5x p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a) = 27a3 + 4(9)a2 – 15a = 27a3 + 36a2 – 15a

c. 3p(a) p(x) = x3 + 4x2 – 5x 3[p(a)] 3[p(a)] = 3[a3 + 4a2 – 5a]

b. p(3a) p(x) = x3 + 4x2 – 5x p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a) = 27a3 + 4(9)a2 – 15a = 27a3 + 36a2 – 15a

c. 3p(a) p(x) = x3 + 4x2 – 5x 3[p(a)] 3[p(a)] = 3[a3 + 4a2 – 5a]

= 3(a3)

b. p(3a) p(x) = x3 + 4x2 – 5x p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a) = 27a3 + 4(9)a2 – 15a = 27a3 + 36a2 – 15a

c. 3p(a) p(x) = x3 + 4x2 – 5x 3[p(a)] 3[p(a)] = 3[a3 + 4a2 – 5a]

= 3(a3)

b. p(3a) p(x) = x3 + 4x2 – 5x p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a) = 27a3 + 4(9)a2 – 15a = 27a3 + 36a2 – 15a

c. 3p(a) p(x) = x3 + 4x2 – 5x 3[p(a)] 3[p(a)] = 3[a3 + 4a2 – 5a]

= 3(a3) + 3(4a2)

b. p(3a) p(x) = x3 + 4x2 – 5x p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a) = 27a3 + 4(9)a2 – 15a = 27a3 + 36a2 – 15a

c. 3p(a) p(x) = x3 + 4x2 – 5x 3[p(a)] 3[p(a)] = 3[a3 + 4a2 – 5a]

= 3(a3) + 3(4a2)

b. p(3a) p(x) = x3 + 4x2 – 5x p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a) = 27a3 + 4(9)a2 – 15a = 27a3 + 36a2 – 15a

c. 3p(a) p(x) = x3 + 4x2 – 5x 3[p(a)] 3[p(a)] = 3[a3 + 4a2 – 5a]

= 3(a3) + 3(4a2) – 3(5a)

b. p(3a) p(x) = x3 + 4x2 – 5x p(3a) = (3a)3 + 4(3a)2 – 5(3a)

= (3)3(a)3 + 4(3)2(a)2 – 5(3a) = 27a3 + 4(9)a2 – 15a = 27a3 + 36a2 – 15a

c. 3p(a) p(x) = x3 + 4x2 – 5x 3[p(a)] 3[p(a)] = 3[a3 + 4a2 – 5a]

= 3(a3) + 3(4a2) – 3(5a) = 3a3 + 12a2 – 15a

Ex. 3 For each graph: 1. Describe the end behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 behavior2. 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 degree3. 2 real zeros

Ex. 4 For each graph: 1. Describe the end behavior2. 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. degree3. real zero(s)

Ex. 4 For each graph: 1. Describe the end behavior2. 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 degree3. 1 real zero