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

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

Practice makes perfect

We want to graph the given polynomial function. f(x)=(x+2)^2(x+4)^2 Let's start by finding the zeros of the function.

Zeros of the Function

To do so, we need to find the values of x for which f(x)=0. f(x)=0 ⇔ (x+2)^2(x+4)^2=0 Since the function is already written in factored form, we will use the Zero Product Property.
(x+2)^2(x+4)^2=0
lc(x+2)^2=0 & (I) (x+4)^2=0 & (II)
lx+2=0 (x+4)^2=0
lx=- 2 (x+4)^2=0
lx=- 2 x+4=0
lx=- 2 x=-4
We found that the zeros of the function are - 2 and - 4.

Graph

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+2)^2(x+4)^2 f(x)=(x+2)^2(x+4)^2
- 5 ( - 5+2)^2( - 5+4)^2 9
- 3 ( - 3+2)^2( - 3+4)^2 1
- 1 ( - 1+2)^2( - 1+4)^2 9
The points ( - 5, 9), ( - 3, 1), and ( -1, 9) 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.
f(x)=(x+2)^2(x+4)^2
f(x)=(x^2+2x(2)+2^2)(x^2+2x(4)+4^2)
f(x)=(x^2+4x+4)(x^2+8x+16)
f(x)=x^2(x^2+8x+16)+4x(x^2+8x+16)+4(x^2+8x+16)
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Distribute x^2 & 8x & 16
f(x)=x^4+8x^3+16x^2+4x(x^2+8x+16)+4(x^2+8x+16)
f(x)=x^4+8x^3+16x^2+4x^3+32x^2+64x+4(x^2+8x+16)
f(x)=x^4+8x^3+16x^2+4x^3+32x^2+64x+4x^2+32x+64
f(x)=x^4+12x^3+52x^2+96x+64
We can see now that the leading coefficient is 1, which is a positive number. Also, the degree is 4, which is an even number. Therefore, the end behavior is up and up. Now, let's draw the graph!