Find a possible formula for a fourth degree polynomial 𝑔 that has a double zero at −1 , 𝑔(−4)=0 , 𝑔(5)=0 , and 𝑔(0)=6.

Question

im so lost. please help. ive been trying to do this problem for a long time.

Answer 1

Hi Brittany! The standards for the State of Virginia tell us everything we need to know to solve this problem: "Given a polynomial function f(x), the following statements are equivalent for any real number, k, such that f(k) = 0:

- k is a zero of the polynomial function f(x), located at (k, 0);

- (x – k) is a factor of f(x);

- k is a solution or root of the polynomial equation f(x) =0" (http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/2016/cf/algebra1-cf.pdf)

So since -1 is a zero, (x - (-1)) is a factor, and we're told we have two (x - (-1)) factors.

Since g(-4) = 0 g(5)=0, -4 and 5 are zeroes too, which gives us two more factors of g(x): (x - (-4)) and (x - 5).

We could have a constant too, call that c. So g(x) = c (x+1)^2 (x+4) (x-5), and all that's left for you to do is figure out what number c is using the fact that g(0)=6.