If x – 2 and x –1/2 both are the factors of the polynomial nx^2 – 5x + m, then show that m = n = 2.

Question

If x – 2 and x – \(\frac1{2}\)both are the factors of the polynomial nx² – 5x + m, then show that m = n = 2.

Answer 1

p(x) = nx² – 5x + m 

(x – 2) is a factor of nx² – 5x + m. 

∴ By factor theorem, 

P(2) = 0 

∴ p(x) = nx² – 5x + m 

∴ p(2) = n(2)²– 5(2) + m 

∴ 0 = n(4) – 10 + m 

∴ 4n – 10 + m = 0 …(i) 

Also, ( x = \(\frac1{2} \)) is a factor of nx² – 5x + m.

∴ By factor theorem,

p( \(\frac1{2} \)) = 0 

p(x) = nx²– 5x + m 

∴ p(\(\frac1{2} \) ) = n(\(\frac1{2} \))² – 5\(\frac1{2} \) + m 

0 = \(\frac{n}{4} \)–\(\frac{5}{2} \) + m 

∴ 0 = n- 10 +4m … [Multiplying both sides by 4] 

∴ n = 10 – 4m ……(ii)

 Substituting n = 10 – 4m in equation (i)

 4(10 – 4m) – 10 + m = 0 

∴ 40 – 16m – 10 + m = 0 

∴ -15m+ 30 = 0 

∴ -15m = -30 

∴ m = 2 Substituting m = 2 in equation (ii), 

n = 10 – 4(2)

 = 10 – 8 

∴ n = 2 

∴ m = n = 2