Question

prism whose size is dependent on the size of the cube.
the volume of the prism is modeled by function $v$.
$v(x) = x^3 - 12x^2 + 35x$
use the x-intercepts to write the factored form of the equation representing the volume
$v(x) =$

Explanation:

Step1: Identify x-intercepts from graph

The x-intercepts are $x=0$, $x=5$, $x=7$.

Step2: Write factored form template

For roots $r_1, r_2, r_3$, factored form is $v(x)=ax(x-r_2)(x-r_3)$

Step3: Find leading coefficient $a$

Compare $ax(x-5)(x-7)$ to original $x^3-12x^2+35x$. Expand factored form:
$$ax(x^2-12x+35)=ax^3-12ax^2+35ax$$
Match to $x^3-12x^2+35x$, so $a=1$.

Step4: Write final factored form

Substitute $a=1$ into the template.

Answer:

$v(x)=x(x-5)(x-7)$