If both (x – 2) and are factors of px² + 5x + r, show that p = r.

Let f(x) = px² + 5x + r

By Factor Theorem, we know that,

If p(x) is a polynomial and a is any real number, then g(x) = (x– a) is a factor of p(x), if p(a) = 0 and vice versa.

So, if (x – 2) is a factor of f(x)

⇒ f(2) = 0

⇒ p(2)² + 5(2) + r = 0

⇒ 4p + r = –10 -------- (A)

Also as is also a factor,

⇒ p + 4r = –10 ----- (B)

On solving equations (A) and (B),we get,

p = r = –2

∴ It is showed that p = r