Question

QUESTION 14The polynomial function y = p(x) has roots at x = -3,x=2, and x = 5, and its y-intercept is y=-6. Write a formula (in factored form) for p(x)

Answer 1

SOLUTION

From the roots of the eqution

$\begin{gathered} x=-3 \\ x=2 \\ x=5 \end{gathered}$

We have

$(x+3)(x-2)(x-5)$

So the form should be

$\begin{gathered} y=a(x+3)(x-2)(x-5)_{} \\ \\ \text{Where a is a constant } \end{gathered}$

Now, let's find a. From

$\begin{gathered} y=a(x+3)(x-2)(x-5) \\ \\ 3\times-2\times-5=30 \end{gathered}$

Since y = -6, we have that

$\begin{gathered} -6=30\times a \\ -6=30a \\ a=\frac{-6}{30} \\ \\ a=-\frac{1}{5} \end{gathered}$

The formula for p(x) becomes

$y=-\frac{1}{5}(x+3)(x-2)(x-5)$