Question

If both (x – 2) and $\left(x-\frac{1}{2}\right)$ are factors of px² + 5x + r, prove that p = r.

Answer 1

Let f(x) = px² + 5x + r

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

Using factor theorem, we have

$$\begin{gathered} f\left(2\right)=0 \\ \Rightarrow p\times 2^2+5\times 2+r=0 \\ \Rightarrow 4p+r=-10.....\left(1\right) \end{gathered}$$
Also, $\left(x-\frac{1}{2}\right)$ is a factor of f(x).

Using factor theorem, we have

$$\begin{gathered} f\left(\frac{1}{2}\right)=0 \\ \Rightarrow p\times {\left(\frac{1}{2}\right)}^2+5\times \frac{1}{2}+r=0 \\ \Rightarrow \frac{p}{4}+r=-\frac{5}{2} \\ \Rightarrow p+4r=-10.....\left(2\right) \end{gathered}$$
From (1) and (2), we have

$$\begin{gathered} 4p+r=p+4r \\ \Rightarrow 4p-p=4r-r \\ \Rightarrow 3p=3r \\ \Rightarrow p=r \end{gathered}$$