Question

The graph of the polynomial function $f\left(x\right)=ax\left(x-4\right)\left(x+1\right)$ passes through the point $\left(5,15\right)$. Re-write the function as the cubic function.

Short answer

The function in cubic notation is written as$f\left(x\right)=2x^3-6x^2-8x$.

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

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

Plugging $a=\frac{1}{2}$in (1),

$f\left(x\right)=2x\left(x-4\right)\left(x+1\right)....\left(2\right)$

To write the function in cubic notation, multiply the respective terms of (2). Therefore, from (2).

$\begin{array}{c}f\left(x\right)=2x\left(x-4\right)\left(x+1\right)\\ =2x^3-6x^2-8x\end{array}$

Step 3. Conclusion.

The function in cubic notation is written as $f\left(x\right)=2x^3-6x^2-8x$.