Question

the function r(x) is defined by the expression -12x² + 72x - 4x³
a) use factoring to determine the zeros of r(x).
b) explain what the zeros represent on the graph of r(x).
c) assess the validity of your answers.

Explanation:

Step1: Rearrange polynomial terms

$r(x) = -4x^3 -12x^2 +72x$

Step2: Factor out GCF

$r(x) = -4x(x^2 + 3x - 18)$

Step3: Factor quadratic trinomial

$r(x) = -4x(x+6)(x-3)$

Step4: Solve for zeros

Set $r(x)=0$:
$-4x=0 \implies x=0$
$x+6=0 \implies x=-6$
$x-3=0 \implies x=3$

Step5: Explain zero graph meaning

Zeros are x-intercepts, where graph crosses x-axis.

Step6: Verify zeros by substitution

For $x=0$: $r(0)=-4(0)^3-12(0)^2+72(0)=0$
For $x=-6$: $r(-6)=-4(-6)^3-12(-6)^2+72(-6)=864-432-432=0$
For $x=3$: $r(3)=-4(3)^3-12(3)^2+72(3)=-108-108+216=0$

Answer:

a) The zeros of $r(x)$ are $x=0$, $x=-6$, and $x=3$.
b) The zeros represent the x-intercepts of the graph of $r(x)$; these are the points where the graph crosses or touches the x-axis, with coordinates $(0,0)$, $(-6,0)$, and $(3,0)$.
c) Substituting each zero back into the original function confirms that $r(x)=0$ for each value, so the zeros are valid.