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Question
The polynomials ax3 + 3x2 − 3 and 2x3 − 5x + a when divided by (x − 4) leave the remainders R1 and R2 respectively. Find the values of the following case, if R1 + R2 = 0.
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Solution
Let us denote the given polynomials as
f(x) = ax3 + 3x2 - 3
g(x) = 2x3 - 5x + a
h(x) = x - 4
Now, we will find the remainders R1 and R2 when f(x) and g(x) respectively are divided by h(x).
By the remainder theorem, when f(x) is divided by h(x) the remainder is
R1 = f(4)
= a(4)3 + 3(4)2 - 3
= 64a + 48 - 3
= 64a + 45
By the remainder theorem, when g(x) is divided by h(x) the remainder is
R2 = g(4)
2(4)3 - 5(4) + a
128 - 20
a + 108
By the given condition,
R1 + R2 = 0
⇒ 64 + 45 + 108 = 0
⇒ 65a + 153 = 0
⇒ 65a = -153
⇒ a = `-153/65`
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