2. If both (x-2) and (x-1/2 ) are factors of px2+5x+r then prove that p=r.
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Answer: As both are the factors of the given polynomials, both must be zero for x = 2 and 1/2. Using factor theorem:
If x - 2 is factor: f(2) = 0
⇒ p(2)² + 5(2) + r = 0
⇒ 4p + 10 + r = 0 ...(1)
If x - 1/2 is factor: f(1/2) = 0
⇒ p(1/2)² + 5(1/2) + r = 0
⇒ p/4 + 5/2 + r = 0
⇒ p + 10 + 4r = 0 ...(2)
Subtract (1) from (2), we get:
⇒ 3p - 3r = 0
⇒ 3p = 3r
⇒ p = r proved
Step-by-step explanation:
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