1. 5.2 Polynomials and Linear Factors
(Day 1)

2. Prime Factors vs. Linear Factors12
Standard Form
Factored Form
(Linear Factors)
Linear Factors are similar to
Prime Factors in that they cannot be ________ any further.
factored

3. Writing a Polynomial in Standard Form
Ex 1.

4. Writing a Polynomial in Factored Form
Ex 2.
Ex 3.
GCF!!!

5. 3x(x2 – 6x + 8) 3x(x – 4)(x – 2) x(x2 – 7x – 18) x(x + 2)(x – 9)
Write each polynomial in factored form. Check by multiplication.
White Boards
4. 3x3 – 18x2 + 24x
5. x3 – 7x2 – 18x
3x(x2 – 6x + 8)
3x(x – 4)(x – 2)
x(x2 – 7x – 18)
x(x + 2)(x – 9)

6. Finding Zeros of a Polynomial Function
6. Find the zeros of y = (x + 1)(x – 1)(x + 3). Then the graph of the function. (Include all min/max points) Use the zero-product property. x = -1, 1, -3 Max Min Graph!
(-2.2, 3.1)
(0.2, -3.1)

7. 7. y = 2x(x + 2)(x – 3) 8. y = –(x + 4)(x + 1)
Find the zeros of each function. Then graph the function.
(Make sure you include the min/max points.)
White Boards
7. y = 2x(x + 2)(x – 3)
8. y = –(x + 4)(x + 1)
Find the relative maximum,relative minimum, and the zeros of each function.
9. f(x) = –2x3 –12x2+40x–12
Together 1st

8. Theorem Zeros = a, b, - c Polynomial Function y = (x – a) (x – b)
The expression x – a is a linear of a polynomial if and only if the value of a is a zero of the related polynomial function.
Factor Theorem:
Zeros =
a, b, - c
Polynomial Function
y =
(x – a)
(x – b)
(x + c)

9. Writing a Polynomial Function From its Zeros
1. Write a polynomial function in standard form with zeros at 2, -3, and 0.
f(x) = (x – 2)(x + 3)x
y = (x2 + 3x – 2x – 6)x
y = (x2 + x – 6)x
y = x3 + x2 – 6x

10. Write a polynomial function in standard form with the given zeros.
White Boards
2. x = 0, 0, 5
3. x = 2, -2, 4

11. Terms Multiple Zero: Multiplicity:
number of times the multiple zero occurs
a repeated zero

12. Finding the Multiplicity of a Zero
4. Find any multiple zeros of f(x) = x5 – 6x4 + 9x3 and state the multiplicity.
f(x) = x5 – 6x4 + 9x3
= x3(x2 – 6x + 9)
= x3(x – 3)(x – 3)
0 with multiplicity
3 with multiplicity
Zeros: 3
odd multiplicity: pass through the x-axis 2
even multiplicity: does NOT pass through the x-axis, it doubles and goes back in the direction it came from
Look at the graph.
(Explain the difference between odd and even multiplicity.)

13. 5. y = 9x5 – 36x3 6. y = (x + 2)4(x – 3) White Boards
Find the zeros of each function. State the multiplicity of multiple zeros.
White Boards
5. y = 9x5 – 36x3
6. y = (x + 2)4(x – 3)

14. Terms is the biggest y-value among the nearby points on the graph
is the least y-value among the nearby points on a graph
Relative Maximum:
Relative Minimum:

15. What is this a picture of?
What part of the graph is important?
Discuss minimum, maximum, x-intercepts (zeros)

16. Summary 1. -4 is a solution of x2 + 3x – 4 = 0
is an x-intercept of the graph of y = x2 + 3x - 4
is a zero of y = x2 + 3x – 4 (also called a root)
x+4 is a factor of x2 + 3x - 4