Question

The polynomial of degree 5, P(x) has leading coefficient 1, has roots of multiplicity 2 at x= 1 andx=0, and a root of multiplicity 1 at x= - 3Find a possible formula for P(x).

Answer 1

Since the leading coefficient of P(x) is 1, and has roots of multiplicity 2 at x=1 and x=0, and a root of multiplicity 1 at x=-3 we get that:

$P(x)=(x-1)^2(x-0)^2(x-(-3))^{}\text{.}$

Simplifying the above polynomial we get:

$\begin{gathered} P(x)=x^2(x^2-2x+1)(x^{}+3) \\ =(x^3+3x^2)(x^2-2x+1)=(x^3+3x^2)x^2-2(x^3+3x^2)x+(x^3+3x^2) \\ =x^5+x^4-5x^3+3x^2\text{.} \end{gathered}$

Answer: A formula for P(x) is:

$P(x)=x^5+x^4-5x^3+3x^2\text{.}$