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.

Answer 1

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`