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.

If the polynomials ax³ + 3x² − 3 and 2x³ − 5x + a, when divided by (x − 4), leave the remainders R₁ and R₂ respectively, find the value of a. It is given that 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 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

`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`