Question

the graph of the function ( p(x) ) has the following features: - passes through the ( x )-axis at ( x = -9 ) and ( x = 0 ) - has multiplicity of 2 at ( x = 4 ) write a possible factored - form equation for ( p(x) ). ( p(x)=x(x + 9)(x - 4)^{2} ) write a different possible factored - form equation for ( p(x) ).

Explanation:

Step1: Recall polynomial root rules

For a polynomial, if $x=a$ is a root, $(x-a)$ is a factor. A root with multiplicity $n$ has $(x-a)^n$ as a factor. A non-zero constant multiplier creates a valid distinct polynomial.

Step2: Choose a constant multiplier

Select any non-zero constant, e.g., $2$.

Step3: Build the new polynomial

Multiply the given polynomial by the constant.
$p(x)=2x(x+9)(x-4)^2$

Answer:

$p(x)=2x(x+9)(x-4)^2$
(Any non-zero constant multiplier works, e.g., $p(x)=-3x(x+9)(x-4)^2$ is also valid)