1. 6-2 Polynomials & Linear Factors
M11.D.2.2.2: Factor Expressions, including differences of squares and trinomials

2. Objectives
The Factored Form of a Polynomial Factors & Zeros of a Polynomial Function

3. Writing a Polynomial in Standard Form
Write (x – 1)(x + 3)(x + 4) as a polynomial in standard form.
(x – 1)(x + 3)(x + 4) = (x – 1)(x2 + 4x + 3x + 12) Multiply (x + 3)
and (x + 4).
= (x – 1)(x2 + 7x + 12) Simplify.
= x(x2 + 7x + 12) – 1(x2 + 7x + 12) Distributive Property
= x3 + 7x2 + 12x – x2 – 7x – 12 Multiply.
= x3 + 6x2 + 5x – 12 Simplify.
The expression (x – 1)(x + 3)(x + 4) is the factored form
of x3 + 6x2 + 5x – 12.

4. Writing a Polynomial in Factored Form
Write 3x3 – 18x2 + 24x in factored form.
3x3 – 18x2 + 24x = 3x(x2 – 6x + 8) Factor out the GCF, 3x.
= 3x(x – 4)(x – 2) Factor x2 – 6x + 8.
Check: 3x(x – 4)(x – 2) = 3x(x2 – 6x + 8) Multiply (x – 4)(x – 2).
= 3x3 – 18x2 + 24x Distributive Property

5. Vocabulary
A relative maximum is the greatest y-value of the points in a region of the graph.
A relative minimum is the smallest y-value of the points in a region of the graph.
Relative Maximum
Relative Minimum

6. Finding Relative Minimum and Maximum Values
Using a graphing calculator, find the relative maximum, relative minimum, and zeros of each function.
x³ + 6x² + 9x + 1
Step 1: Go to y= and enter the function and then graph
Step 2: Hit 2nd Calc : Maximum
Left Bound? : Move your cursor to the left of the “peak” and hit enter
Right Bound?: Move your cursor to the right of the “peak” and hit enter
Guess? : Just hit enter again
Step 3: Hit 2nd Calc : Minimum
Follow the same steps as 2
Step 4: Hit 2nd Calc : Zero
Follow the same steps as 2, but your cursor needs to be to the left of where the function intersects the x-axis.

7. Finding Zeros of a Polynomial Function
Find the zeros of y = (x + 1)(x – 1)(x + 3). Then graph the function using a graphing calculator.
Using the Zero Product Property, find a zero for each linear factor.
x + 1 = 0    or     x – 1 = 0    or    x + 3 = 0
x = –1 x = x = –3
The zeros of the function are –1, 1, –3.
Now sketch and label the function.

8. Writing a Polynomial Function From its Zeros
Write a polynomial in standard form with zeros at 2, –3, and 0.
2 –3 0 Zeros
ƒ(x) = (x – 2)(x + 3)(x) Write a linear factor for each zero.
= (x – 2)(x2 + 3x) Multiply (x + 3)(x).
= x(x2 + 3x) – 2(x2 + 3x) Distributive Property
= x3 + 3x2 – 2x2 – 6x Multiply.
= x3 + x2 – 6x Simplify.
The function ƒ(x) = x3 + x2 – 6x has zeros at 2, –3, and 0.

9. In the example above, the multiplicity is 2
Vocabulary
Let’s look at an example:
has zeros at -2, 3, & 3.
If a linear factor of a polynomial is repeated, it is called a multiple zero.
A multiple zero has a multiplicity equal to the number of times a zero occurs.
In the example above, the multiplicity is 2

10. Finding the Multiplicity of a Zero
Find any multiple zeros of ƒ(x) = x5 – 6x4 + 9x3 and state the multiplicity.
ƒ(x) = x5 – 6x4 + 9x3
ƒ(x) = x3(x2 – 6x + 9) Factor out the GCF, x3.
ƒ(x) = x3(x – 3)(x – 3) Factor x2 – 6x + 9.
Since you can rewrite x3 as (x – 0)(x – 0)(x – 0), or
(x – 0)3, the number 0 is a multiple zero of the function, with multiplicity 3.
Since you can rewrite (x – 3)(x – 3) as (x – 3)2, the number 3 is a multiple
zero of the function with multiplicity 2.

11. Homework
Pg 317 # 1, 2, 8, 10, 13, 21, 22, 29