Question

if the polynomials (3x³+ax²+3x+5) and (4x³+x²-2x+a) leave the same remainder when divided by (x-2), find the value of a. Also find them remainder in each 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

Answer 2

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