1. 5.2 Polynomials, Linear
Factors, and Zeros

2. Writing a Polynomial in Factored
Form
 In Chapter 4, we solved quadratic
functions by factoring and setting each
factor equal to zero
 We can solve some polynomial
functions in a similar way.
◦ Remember: ALWAYS factor out the GCF
first!

3. Linear Factors, Roots, Zeros,
and x-intercepts
 The following are equivalent
statements about a real number b and
a polynomial P(x)
◦ (x – b) is a linear factor of the polynomial
P(x)
◦ b is a zero of the polynomial function y =
P(x)
◦ b is a root (or solution) of the polynomial
equation P(x) = 0
◦ b is an x-intercept of the graph of y =
P(x)

4. Example: Write each polynomial
in factored form. Then, find the
zeros of the function.

5. Example: Write each polynomial
in factored form. Then, find the
zeros of the function.

6. Graphing a Polynomial
Function
1. Find the zeros and plot them
2. Find points between the zeros and
plot them
3. Determine the end behavior
4. Sketch the graph

7. Example: Find the zeros of the
function. Then graph the
function.

8. Example: Find the zeros of the
function. Then graph the
function.

9. The Factor Theorem
 The factor theorem describes the
relationship between the linear factors
of a polynomial and the zeros of a
polynomial.
 The Factor Theorem
The expression x – a is a factor of a
polynomial if and only if the value a is
a zero of the related polynomial
function

10. Using the Factor Theorem to
Write a Polynomial
1. Write each zero as a factor
2. Multiply and combine like terms

11. Example: Write a polynomial
function in standard form with the
given zeros
X = –2, 2, and 3

12. Example: Write a polynomial
function in standard form with the
given zeros
X = –2, –2, 2 and 3

13. Homework
 P293 #7 – 26, 61 – 63

14. Multiple Zeros and Multiplicity
 A multiple zero is a linear factor that
is repeated when the polynomial is
factored completely
 The multiplicity of a zero is the
number of times the linear factor is
repeated in the factored form of the
polynomial.
◦ If a zero is of even multiplicity, then the
graph touches the x-axis and “turns
around”
◦ If a zero is of odd multiplicity, then the
graph crosses the x-axis

15. Example: Find the zeros of the
function. State the multiplicity of
multiple zeros.
What does the
multiplicity tell
you about the
graph?