Question

If $x-2$ and $x-\frac{1}{2}$ both are the factors of the polynomial nx² − 5x + m, then show that $m=n=2$

Answer 1

Let p(x) = nx² − 5x + m.

It given that (x − 2) and $\left(x-\frac{1}{2}\right)$ are the factors of the polynomial p(x) = nx² − 5x + m.

∴ By factor theorem, p(2) = 0 and $p\left(\frac{1}{2}\right)=0$.

$$\begin{gathered} p\left(2\right)=0 \\ \Rightarrow n\times {\left(2\right)}^2-5\times 2+m=0 \\ \Rightarrow 4n-10+m=0 \\ \Rightarrow 4n+m=10.....\left(1\right) \end{gathered}$$
Also,

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

4n + m = n + 4m

⇒ 4n − n = 4m − m

⇒ 3n = 3m

⇒ n = m

Putting n = m in (1), we have

4m + m = 10

⇒ 5m = 10

⇒ m = 2

∴ n = m = 2