Question

The graph of the polynomial function $f\left(x\right)=ax\left(x-4\right)\left(x+1\right)$ passes through the point $\left(5,15\right)$. Find the value of $a$.

Short answer

The value of$a=\frac{1}{2}$.

Step-by-step solution

Step 1. Write down the given information.

It is given that the graph of the polynomial function $f\left(x\right)=ax\left(x-4\right)\left(x+1\right)$ passes through the point $\left(5,15\right)$. Therefore, the point $\left(5,15\right)$will satisfy,

$f\left(x\right)=ax\left(x-4\right)\left(x+1\right)....\left(1\right)$

Step 2. Calculation.

Since the graph of the polynomial function $f\left(x\right)=ax\left(x-4\right)\left(x+1\right)$ passes through the point $\left(5,15\right)$. Therefore, the point $\left(5,15\right)$will satisfy $f\left(x\right)=ax\left(x-4\right)\left(x+1\right)$.

$\begin{array}{c}15=a\left(5\right)\left(\left(5\right)-4\right)\left(\left(5\right)+1\right)\\ 15=30a\\ a=\frac{1}{2}\end{array}$

Step 3. Conclusion.

The value of $a=\frac{1}{2}$.