Question

If both (x - 2) and $(x-\frac{1}{2})$ are factors of $p{x}^2+5x+r$, prove that p = r.

Answer 1

Let f(x) = $p{x}^2+5x+r$

It is given that (x – 2) is a factor of f(x).

Using factor theorem, we have

f(2)=0
⇒$p(2{)}^2+5(2)+r=0$
⇒4p+r=-10 .....(1)

Also, $(x-\frac{1}{2})$ is a factor of f(x).
Using factor theorem, we have
$f\frac{1}{2}=0$
⇒$p\times {\frac{1}{2}}^2+5\times \frac{1}{2}+r=0$
⇒$\frac{p}{4}+r=\frac{-5}{2}$
⇒p+4r=-10 .....(2)
From (1) and (2), we have
4p+r=p+4r
⇒4p-p=4r-r
⇒3p=3r
⇒p=r