write an equation for a polynomial function whose graph intercepts the horizontal axis at -7, 8, and 15.

Explanation:

Step1: Recall zero - root relationship

If a polynomial function $P(x)$ has a root $r$, then $(x - r)$ is a factor of $P(x)$. The roots of the polynomial are the $x$-values where the graph intercepts the horizontal axis. Here, the roots are $x=-7$, $x = 8$, and $x=15$.

Step2: Construct the polynomial

The polynomial function can be written as $P(x)=a(x + 7)(x - 8)(x - 15)$, where $a$ is a non - zero constant. For simplicity, we can take $a = 1$. So $P(x)=(x + 7)(x - 8)(x - 15)$.

Step3: Expand the polynomial

First, expand $(x + 7)(x - 8)$: [ \begin{align*} (x + 7)(x - 8)&=x^2-8x+7x - 56\ &=x^2-x - 56 \end{align*} ] Then multiply by $(x - 15)$: [ \begin{align*} P(x)&=(x^2-x - 56)(x - 15)\ &=x^3-15x^2-x^2 + 15x-56x+840\ &=x^3-16x^2-41x + 840 \end{align*} ]

Answer:

$P(x)=(x + 7)(x - 8)(x - 15)$ (or $P(x)=x^3-16x^2-41x + 840$)