Math

QuestionAnalyze the polynomial f(x)=(x+3)(x5)(x+6)f(x)=(x+3)(x-5)(x+6).
(a) What is the end behavior? (b) Find the xx- and yy-intercepts.

Studdy Solution

STEP 1

Assumptions1. The polynomial function is given by f(x)=(x+3)(x5)(x+6)f(x)=(x+3)(x-5)(x+6). We need to analyze the function by determining the end behavior, x-intercepts and y-intercepts

STEP 2

We start by determining the end behavior of the function. The end behavior of a function refers to the behavior of the function as xx approaches positive infinity (xx \rightarrow \infty) and negative infinity (xx \rightarrow -\infty).For a polynomial, the end behavior is determined by the degree and the leading coefficient of the polynomial.

STEP 3

First, we need to find the degree of the polynomial. The degree of a polynomial is the highest power of xx in the polynomial.In this case, the polynomial is in factored form. The degree of the polynomial is the sum of the powers of xx in each factor.Since each factor is a linear term (power of1), the degree of the polynomial is 1+1+1=31+1+1=3.

STEP 4

Next, we need to find the leading coefficient of the polynomial. The leading coefficient is the coefficient of the term with the highest power of xx.To find the leading coefficient, we need to expand the polynomial and look at the coefficient of the x3x^3 term.However, since the coefficients of xx in each factor are all1, the leading coefficient of the polynomial is 111=11*1*1=1.

STEP 5

Now that we have the degree and the leading coefficient of the polynomial, we can determine the end behavior.For a polynomial of odd degree and positive leading coefficient, as xx \rightarrow \infty, f(x)f(x) \rightarrow \infty and as xx \rightarrow -\infty, f(x)f(x) \rightarrow -\infty.

STEP 6

Next, we find the x-intercepts of the function. The x-intercepts are the values of xx where the function equals zero, i.e., f(x)=0f(x)=0.
Setting each factor in the polynomial equal to zero gives the x-intercepts.

STEP 7

olve the equations x+3=0x+3=0, x5=0x-5=0 and x+6=0x+6=0 to find the x-intercepts.

STEP 8

The solutions to these equations are x=3x=-3, x=5x=5 and x=6x=-6. These are the x-intercepts of the function.

STEP 9

Finally, we find the y-intercept of the function. The y-intercept is the value of the function when x=x=, i.e., f()f().
Substitute x=x= into the function to find the y-intercept.

STEP 10

Calculate f(0)=(0+3)(05)(0+6)f(0)=(0+3)(0-5)(0+6).

STEP 11

The result is f(0)=3(5)6=90f(0)=3*(-5)*6=-90. This is the y-intercept of the function.

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