The polynomial function P(x) of degree 5 has a leading coefficient of 1, Zeros of multiplicity 2 at x=5x=5 and x=0x=0, and a Zero of multiplicity 1 at x=−3x=-3

Question

The polynomial function P(x) of degree 5 has a leading coefficient of 1, Zeros of multiplicity 2 at x=5 and x=0, and a Zero of multiplicity 1 at x=−3

Find the polynomial function P(x). Leave your answer in factored form.

P(x)= ___________________

Answer 1

P(x) = x^2(x-5)^2(x+3)

multiplicity tells you the exponent of the factor

change the sign of the zero and stick an x in front of it to get the factor

leading coefficient of 1 doesn't change anything. degree 5 is x^2(x^2)x = x^5 with implicit 1 as its coefficient

P(x) = x^2(x^2-10x+25)(x+3)

= x^2(x^3 -7x^2 -5x +75)

=x^5 -7x^4 -5x^2 +75x^2 has leading coefficient of 1, the coefficient of the 5th degree term