Polynomials help (1 Viewer)

daddywal · Sep 8, 2023

Q.

I am getting alpha = -5 but the answers say alpha =11/9. I have tried via long division and by inspection, but I am still getting the same answer.

sunmoonlight · Sep 8, 2023

The answer is -5

Luukas · Sep 17, 2023

Applying the factor theorem for the root at $x=-2$ :

$\begin{align*} p(-2) &= a(-2)^3 + 16(-2)^2 + c(-2) - 120 \\ 0 &= -8a - 2c - 56 \\ 4a + c &= -28 \end{align*}$

Applying the factor theorem for the root at $x=3$ :

$\begin{align*} p(3) &= a(3)^3 + 16(3)^2 + c(3) - 120 \\ 0 &= 27a + 3c + 24 \\ 9a + c &= -8\end{align*}$

The difference between these equations is

$\begin{align*} (9a + c) - (4a + c) &= -8 - -28 \\ 5a &= 20 \\ a &= 4 \end{align*}$

and so the sum of the roots is

$\begin{align*} -2 + 3 + \alpha &= \frac{-16}{4} = -4 \\ \alpha &= -5 \end{align*}$ .