Math

QuestionAnalyze the polynomial f(x)=(x+1)(x2)(x+5)f(x)=(x+1)(x-2)(x+5): end behavior, intercepts, and zeros. Find xx-intercepts: -1, 2, -5; yy-intercept: -10.

Studdy Solution

STEP 1

Assumptions1. The polynomial function is f(x)=(x+1)(x)(x+5)f(x)=(x+1)(x-)(x+5). We are asked to analyze the function using parts (a) through (e)

STEP 2

(a) Determine the end behavior of the graph of the function. The degree of the polynomial is (since it is a product of three linear factors), and the leading coefficient is positive (since the coefficient of xx in each factor is positive). Hence, the end behavior of the function is the same as that of y=x^.

STEP 3

(b) Find the xx - and yy-intercepts of the graph of the function. The xx-intercepts of the function are the solutions to the equation f(x)=0f(x)=0, which are the roots of the polynomial. These are the values of xx that make each factor equal to zero.

STEP 4

Set each factor equal to zero and solve for xx to find the xx-intercepts.
x+1=0,x2=0,x+=0x+1=0, x-2=0, x+=0

STEP 5

olving the above equations gives the xx-intercepts as x=1,2,5x=-1,2, -5.

STEP 6

The yy-intercept of the function is the value of f(x)f(x) when x=0x=0. Substitute x=0x=0 into the function to find the yy-intercept.

STEP 7

Substitute x=0x=0 into the function f(x)=(x+1)(x2)(x+5)f(x)=(x+1)(x-2)(x+5).
f(0)=(0+1)(02)(0+5)f(0)=(0+1)(0-2)(0+5)

STEP 8

Calculate the yy-intercept.
f(0)=1×2×5=10f(0)=1 \times -2 \times5 = -10The yy-intercept is 10-10.

STEP 9

(c) Determine the zeros of the function and their multiplicity. The zeros of the function are the same as the xx-intercepts, which are ,2,5-,2, -5.

STEP 10

The multiplicity of a zero is the number of times it appears as a root. In this case, each zero appears once, so each has multiplicity.

STEP 11

Use this information to determine whether the graph crosses or touches the x\mathrm{x}-axis at each x\mathrm{x}-intercept. Since each zero has multiplicity (which is odd), the graph crosses the x\mathrm{x}-axis at each x\mathrm{x}-intercept.

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