Big Ideas Math Algebra 2, 2014
BI
Big Ideas Math Algebra 2, 2014 View details
8. Analyzing Graphs of Polynomial Functions
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Exercise 5 Page 216

Use the Zero Product Property to find the zeros of the polynomial function.

B

Practice makes perfect
We want to find the zeros of the given polynomial function. g(x)=(x+1)(x-1)(x+2)

Zeros of the Function

To do so, we need to find the values of x for which g(x)=0. g(x)=0 ⇔ (x+1)(x-1)(x+2)=0 Since the function is already written in factored form, we will use the Zero Product Property.
(x+1)(x-1)(x+2)=0
lcx+1=0 & (I) x-1=0 & (II) x+2=0 & (III)
lx=- 1 x-1=0 x+2=0
lx=- 1 x=1 x+2=0
lx=- 1 x=1 x=- 2
We found that the zeros of the function are - 2, - 1, and 1. We can stop here and recognize that the only graph with those zeros is graph B. We can also continue by graphing the function with a table of values as shown below.

Extra

Graphing the function using a table of values

To draw the graph of the function, we will find some additional points and consider the end behavior. Let's use a table to find additional points.

x (x+1)(x-1)(x+2) g(x)=(x+1)(x-1)(x+2)
- 3 ( - 3+1)( - 3-1)( - 3+2) - 8
- 1.5 ( - 1.5+1)( - 1.5-1)( - 1.5+2) 0.625
0 ( 0+1)( 0-1)( 0+2) - 2
0.5 ( 0.5+1)( 0.5-1)( 0.5+2) - 1.875
The points ( - 3, - 8), ( - 1.5, 0.625), ( 0, - 2), and ( 0.5, - 1.875) are on the graph of the function. Finally, let's apply the Distributive Property. This will simplify the equation and determine the leading coefficient and degree of the polynomial function.
g(x)=(x+1)(x-1)(x+2)
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Multiply
g(x)=(x(x-1)+1(x-1))(x+2)
g(x)=(x^2-x+1(x-1))(x+2)
g(x)=(x^2-x+x-1)(x+2)
g(x)=(x^2-1)(x+2)
g(x)=x^2(x+2)-1(x+2)
g(x)=x^3+2x^2-1(x+2)
g(x)=x^3+2x^2-x-2
We can see now that the leading coefficient is 1, which is a positive number. Also, the degree is 3, which is an odd number. Therefore, the end behavior is down and up. Now, let's draw the graph!