ALGEBRA 25.3 Day 1

Polynomial Functions

RECALL

A polynomial is an expression that is a sum of variables and exponents.

Degree Type Example0 Constant 121 Linear2 Quadratic3 Cubic4 Quartic

Degree n Examples vary

LEADING COEFFICIENT

The coefficient of the first term of a polynomial in standard form is called the leading coefficient.

Example Leading coefficient12 12

45

8

EXAMPLE 1

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why.a) b)

c)

REAL WORLD – EXAMPLE 2

RESPIRATION The volume of air in the lungs during a 5-second respiratory cycle can be modeled by v(t) = –0.037t

3 + 0.152t

2 + 0.173t, where v is the volume in liters and t is the time in seconds. This model is an example of a polynomial function. Find the volume of air in the lungs 1.5 seconds into the respiratory cycle.

EXAMPLE 3

If , find .Step 1: Find

Step 2: Find

Step 3: Now subtract step 1 and 2.

YOU TRY

Find g(2x + 1) – 2g(x) if g(b) = b2 + 3.

A. 1

B. 2x

2 + 4x – 2

C. 2x

2 + 4x + 10

D. 2x

2 – 2

Hint:Step 1: Find Step 2: Find Step 3: Now subtract step 1 and 2.

EXIT SLIP

1. Determine whether 3x3 + 2x2 – 3 is a polynomial in one variable. If so, state the degree and leading coefficient.

2. Find if

ALGEBRA 25.3 Day 2

Polynomial Functions

Zeros of Even- and Odd-Degree Functions Odd-degree functions will always have an

odd number of real zeros. Even-degree functions will always have an

even number of real zeros or no zeros at all.

The number of turns is always one less than the degree.

EXAMPLE 1For each graph, Describe the end behavior Determine whether it represents an odd-

degree or an even-degree polynomial function

State the number of real zerosa) b)

YOU TRY

For the graph, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros.

EXIT SLIPFor the graph, Describe the end behavior Determine whether it

represents an odd-degree or an even-degree polynomial function

State the number of real zeros