Question

Construct a polynomial function with the following properties: fifth degree, 5 is a zero of multiplicity 4, −4 is the only other zero, leading coefficient is 5

Answer 1

5(x-5)^4(x+4)
you raise x-5 to the 4th power because 5 has a multiplicity of 4

Answer 2

[tex]p(x)=5(x-5)^4(x+4)[\tex]

Explanation:

If an nth order polynomial P(x) has a zero, say w, then P(x) can be written as P(x)=(x-w)Q(x) for some Q(x) , a polynomial of degree (n-1). So, 5 is a zero with multiplicity 4, then {tex]P(x)=(x-5)^4 Q(x) [\tex} for some Q(x). -4 is the only other zero, so (x-(-4))=(x+4) is a factor of Q(x). P(x) is of fifth degree , so we have exhausted the possibilities to [tex]p(x)=(x-5)^4(x+4)R(x)[\tex] where R(x) is of degree 0 or a constant. But the leading coefficient is 5, so is the answer.