Question

if the polynomial (2x^3 + ax^2 =3x - 5) and (x^3 + x^2 - 2x + a) leave the same remainder when divided by (x - 2) find the value of a and remainder in eaxh case

Answer 1

Let the given polynomials be f(x) = 3x3+ax2+3x+5 g(x) = 4x3+x2-2x+a Given that when f(x) and g(x) are divided by the (x - 2) we get the same remainders That is f(2) = g(2) Put x = 2 in both f(x) and g(x) 3(2)3+ a(2)2+ 3(2) + 5 = 4(2)3+ (2)2 - 2(2) + a ⇒ 24 + 4a + 6 + 5 =32 + 4 - 4 + a ⇒ 35 + 4a = 32 + a ⇒ 3a = – 3 ∴ a = – 1 f(x) = 3x3– x2+ 3x + 5 The remainder is f(2) Hence f(2) = 3(2)3– (2)2+ 3(2) + 5                  = 24 – 4 + 6 + 5 = 31