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 cases, if 2R₁ − R₂ = 0.

Answer 1

Let us denote the given polynomials as

`f(x) = ax^3 + 3x^2 -3`

`g(x) = 2x^3 - 5x + a,`

` h(x) = x-4`

Now, we will find the remainders R₁and 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

`R_1 = f(4)`

` = a(4)^3 + 3(4)^2 -3`

` = 64a + 48 - 3`

` = 64a + 48`

By the remainder theorem, when g(x) is divided by h(x) the remainder is

`R_2 = g(4)`

`2(4)^3 - 5(4) + a`

`128 - 20`

` a+108`

 By the given condition,

2R₁ − R₂ = 0

`⇒ 2(64a + 45) - (a+ 108) = 0 `

`⇒              128a + 90 - a - 108 = 0 `

`⇒                                                                      127a = 18`

` ⇒                                                                                 a = 18/127`