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Chapter 4: Q. 39 (page 245)

Use the graph below of a polynomial function y=f(x)to solve

(a)f(x)=0(b)f(x)>0(c)f(x)鈮�0

(d) determine f

Short Answer

Expert verified

(a)The function is 0 at x=-3,2,

(b)f(x)>0atx>-3andx>2,

(c)f(x)鈮�0atx鈮�-3andx=2

(d)f(x)=x3-x2-8x+12

Step by step solution

01

Step 1. Given Information

The given graph of the function is

02

Part(a) Step 1. Explanation

Locate the points on the graph where the function intersects the x-axis.

Thus, atx=-3andx=2the function is equal to 0.

03

Part(b) Step 1. Explanation

Determine the domain of the function for which the function is above the x-axis.

At x>-3andx>2the function is strictly greater than 0.

In the interval notation,x:(-3,鈭�)鈭�(2,鈭�)

04

Part(c) Step 1. Explanation

Determine the domain of the function for which the function is below or intersecting the x-axis.

Atx鈮�-3andx=2and in interval notation the solution isx:(-鈭�,-3]鈭�2

05

Part(d) Step 1. Calculation

Use the zeroes from part (a) to produce a polynomial. Expand the linear factors.

f(x)=(x+3)(x-2)(x-2)=(x+3)(x2-4x+4)=x3-4x2+4x+3x2-12x+12=x3-x2-8x+12

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