Construct a polynomial function with the following properties: fifth degree, 1 is a zero of multiplicity 2, −4 is the only other zero, leading coefficient is 4.

Answer 1

If the polynomial is 5th degree, it must have 5 zeros.

The 1 zero with a multiplicity of 2 accounts for 2 of the 5 required zeros. The "only other zero" would have to have multiplicity of 3 so that we can account for all 5 required zeros.

I will assume that this is the case: here are the 5 zeros:

x=1, x=1, x= -4, x= -4, x=-4

Turn these into factors

x-1 = 0 x-1 = 0, x + 4 = 0, x + 4 = 0, x + 4 = 0

So the polynomial would be (x-1)(x-1)(x+4)(x+4)(x+4) to get all of the correct zeros.

If the leading coefficient is 4, then the polynomial becomes

f(x) = 4 (x-1)² (x+4)³