Question

If the polynomials ax³ + 3x² − 13 and 2x³ − 5x + a, when divided by (x − 2) leave the same remainder, find the value of a.

Answer 1

Let us denote the given polynomials as

`f(x) = ax^3 + 3x^2 - 13`

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

`h(x) = x -2`

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(2)`

    ` = a(2)^3 + 3(2)^2 - 13`

    ` = 8a + 12 - 13`

    ` = 8a - 1`

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

`R_2 = g(2)`

     ` = 2(2)^3 - 5 (2) + a`

     ` = 16 - 10 + a`

     ` = a+6`

By the given condition, the two remainders are same. Then we have, R₁ = R₂

`⇒ 8a - 1 = a+ 6`

`⇒ 8a - a = 6+1`

`⇒              7a = 7`

`⇒               a =7/ 7`

`⇒                 a = 1`

`⇒                  a = 1`