Question

3. Use the given information about the polynomial graph to write out the equation.

Answer 1

Answer:

$f(x)=-\frac{1}{12}(x-3)^2(x+2)^2(x-4)$

Step-by-step explanation:

The standard form for a polynomial equation in its factored form is:

$f(x)=a(x-p)(x-q)...$

Where p and q are the zeros and a is the leading coefficient.

We know that the degree of the polynomial is 5.

We are also given roots at x=3 and x=-2. Hence, our factors are (x-3) and (x+2).

These have a multiplicity of two, so we will square them.

We also know that there is a root at x=4. Hence, the factor is (x-4). This has a multiplicity of 1.

Therefore, our polynomial is now:

$f(x)=a(x-3)^2(x+2)^2(x-4)$

We know that the y-intercept is 12. Therefore, if we substitute in 0 for x, we should get 12 for y. Here, we are solving for our a. Therefore:

$12=a(0-3)^2(0+2)^2(0-4)$

Evaluate:

$12=a(9)(4)(-4)$

Evaluate:

$-144a=12$

Divide both sides by -144:

$a=-1/12$

Hence, our entire polynomial is:

$f(x)=-\frac{1}{12}(x-3)^2(x+2)^2(x-4)$