Question

If both x – 2 and `x - 1/2` are factors of px² + 5x + r, show that p = r.

Answer 1

Let p(x) = px² + 5x + r

Given (x – 2) is a factor of p(x)

So, p(2) = 0

p(2)² + 5 × 2 + r = 0

4p + 10 + r = 0  ...(1)

Again, `(x - 1/2)` is a factor of p(x).

So, `p(1/2)` = 0

Now, `p(1/2) = p(1/2)^2 + 5 xx (1/2) + r`

= `1/4p + 5/2 + r`

`p(1/2) = 0`

`1/4 p + 5/2 + r = 0`

From (1), we have 4p + r = –10

From (2), we have p + 10 + 4r = 0

p + 4r = –10

4p + r = p + 4r

3p = 3r

p = r

Hence, proved.