I Need Help With Writing a Polynomial Function Equation (College Algebra)

Question

The question asks

"Find a degree 4 polynomial having zeros −5, −2, 4 and 5 and the leading coefficient is equal to −2. "

P(x)=???

I need help finding the factored form, not simplified.

I have tried 3 different ways to solve this problem but it keeps telling me I get the answer incorrect. Thank you!

Answer 1

Each zero was once a factor.

x = -5 Add 5 to both sides and get the factor (x+5)=0

x = -2 Add 2 to both sides and get the factor (x+2)=0

x = 4 Add -4 to both sides and get the factor (x-4)=0

x = 5 Add -5 to both sides and get the factor (x-5)=0

f(x) = A(x+5)(x+2)(x-4)(x-5)

And since the leading coefficient is -2, this is what goes in place of "A" in the f(x)

f(x) = -2 (x+5) (x+2) (x-4) (x-5)

Here is graph of our f(x) - ignore the y-values, I had to shrink it to get it to fit, but pay attention to the zeros, the points where it crosses the x-axis. Notice they are the zeros given in the question. So we have the correct function.