warm-up: simplify

Warm-up: Simplify.

222 5

8

993 7

98

7 130 62 4

5 31. 3 3 2. 3. 5 7

6.8 104. (2.5x10 )(1.8x10 ) 5.1.5 10

456. 74 7.9

xx

x yxx y

Warm-up: Answers

27 6

2

222 5

8

993 7

98

7 130 6

10

3

18 2 54

5 31. 3 3 2. 3. 5 7

6.8 104. (2.5x10 )(1.8x10 ) 5.1.5 10

456

3 93 57 49

4.5x10 4.533x10

1. 7 54 7.9

xx

x y yy xxx

x

Check Homework

p. 533 # 15-43 odd

Quiz 5.1 tomorrow

Objective(5.2A): Classify polynomials and evaluate polynomials using synthetic substitution.

A Polynomial Function is a function of the form

• f(x) = anxn + an-1xn-1 + …+a1x + a0

The exponents are all whole numbers, and the coefficients are all real numbers.

an is the leading coefficientn is the degree

Degree Type Standard Form Example

You are already familiar with some types of polynomialfunctions. Here is a summary of common types ofpolynomial functions.

4 Quartic f (x) = 5x 4 + 4 x

3 + 7 x 2 + 9 x + 1

0 Constant f (x) = 4

3 Cubic f (x) = 5 x 3 + 3 x

2 - x + 6

2 Quadratic f (x) = 5 x 2 + 2 x + 7

1 Linear f (x) = 2x -10

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

f (x) = x 2 – 3x4 – 71

2

SOLUTION

The function is a polynomial function.

It has degree 4, so it is a quartic function.

The leading coefficient is – 3.

Its standard form is f (x) = – 3x 4

+ x 2 – 7. 1

2

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

The function is not a polynomial function because the term 3

x does not have a variable base and an exponentthat is a whole number.

SOLUTION

f (x) = x 3 + 3

x

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

SOLUTION

f (x) = 6x 2 + 2 x

–1 + x

The function is not a polynomial function because the term2x

–1 has an exponent that is not a whole number.

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

SOLUTION

The function is a polynomial function.

It has degree 2, so it is a quadratic function.

The leading coefficient is .

Its standard form is f (x) = x2 – 0.5x – 2.

f (x) = – 0.5 x + x 2 – 2

f (x) = x 2 – 3 x

4 – 712

f (x) = x 3 + 3x

f (x) = 6x2 + 2 x– 1 + x

Polynomial function?

f (x) = – 0.5x + x2 – 2

What is the degree of the monomial? 245 bx

The degree of a monomial is the sum of the exponents of the variables in the monomial. The exponents of each variable are 4 and 2. 4+2 = 6.

The degree of the monomial is 6. The monomial can be referred to as a sixth degree monomial.

Classify the polynomials by degree and number of terms.

Polynomiala.

b.

c.

d.

5

42 x

xx 23

14 23 xx

DegreeClassify by

degree

Classify by number of

termsZero Constant Monomial

First Linear Binomial

Second Quadratic Binomial

Third Cubic Trinomial

Write the polynomials in standard form and identify the polynomial by degree and number of terms. 23 237 xx a)

b) xx 231 2

3 23 8 11x x x c)

• Page 338 (1-5)

One way to evaluate polynomial functions is to usedirect substitution. Another way to evaluate a polynomialis to use synthetic substitution.

Use synthetic division to evaluate

f (x) = 2 x 4 + 8 x

2 + 5 x 7 when x = 3.

Just Watch--

Polynomial in standard form

2 x 4 + 0 x

3 + (–8 x 2) + 5 x + (–7)

2 6

6

10

18

35

30 105

98

The value of f (3) is the last number you write,In the bottom right-hand corner.

2 0 –8 5 –7 CoefficientsCoefficients

3

x-value

3 •

SOLUTION

Polynomial instandard form

EXAMPLE 3 Evaluate by synthetic substitution

a) Use synthetic substitution to evaluate f (x) from Example 2 when x = 3. f (x) = 2x4 – 5x3 – 4x + 8

SOLUTION

STEP 1 Write the coefficients of f (x) in order of descending exponents. Write the value at which f (x) is being evaluated to the left.

EXAMPLE 3 Evaluate by synthetic substitution

STEP 2 Bring down the leading coefficient. Multiply the leading coefficient by the x-value. Write the product under the second coefficient. Add.

STEP 3 Multiply the previous sum by the x-value. Write the product under the third coefficient. Add. Repeat for all of the remaining coefficients. The final sum is the value of f(x) at the given x-value.

EXAMPLE 3 Evaluate by synthetic substitution

Synthetic substitution gives f(3) = 23, which matches the result in Example 2.

ANSWER

GUIDED PRACTICE for Examples 3 and 4

Use synthetic substitution to evaluate the polynomial function for the given value of x.

b) f (x) = 5x3 + 3x2 – x + 7; x = 2

Write the coefficients of f (x) in order of descending exponents. Write the value at which f (x) is being evaluated to the left.

GUIDED PRACTICE for Examples 3 and 4

Synthetic substitution gives f(2) = 57ANSWER

GUIDED PRACTICE for Examples 3 and 4

c) g (x) = – 2x4 – x3 + 4x – 5; x = – 1

Write the coefficients of g(x) in order of descending exponents. Write the value at which g (x) is being evaluated to the left.

– 1 – 2 – 1 0 4 – 5

GUIDED PRACTICE for Examples 3 and 4

2 –1 1 –5– 1 – 2 – 1 0 4 – 5

– 2 1 –1 5 – 10

Synthetic substitution gives f(– 1) = – 10ANSWER

Assignments

Classwork: Practice 5.2 # 1-7

Homework (5.2A): p. 341 # 3-8, 15-23 (15 pts)

Closure: Review exponent rules for quiz tomorrow