if both x-2 and x-1/2 are factors of px2+5x+r show that p=r - Maths - Polynomials

Question

if both x-2 and x-1/2 are factors of
px2+5x+r show that p=r

Ajanta Trivedi · Accepted Answer

the given polynomial is $q (x) = p x^{2} + 5 x + r$ .......(1)

if x-2 is the factor of q(x) then q(2) = 0

$p (2)^{2} + 5 * 2 + r = 0 4 p + r + 10 = 0......... (2)$

if x-1/2 is the fctor of q(x) then q(1/2) = 0

$p (\frac{1}{2})^{2} + 5 * \frac{1}{2} + r = 0 \frac{p}{4} + \frac{5}{2} + r = 0 p + 4 r + 10 = 0...... (3)$

subtracting (3) from (2):

$4 p + r + 10 - (p + 4 r + 10) = 0 \Rightarrow 3 p - 3 r = 0 \Rightarrow 3 p = 3 r \Rightarrow p = r$

which is the required result.

hope this helps you.

cheers!!