Question

If the polynomials 2x³ + ax² + 3x − 5 and x³ + x² − 4x +a leave the same remainder when divided by x −2, find the value of a.

Answer 1

Let us denote the given polynomials as

`f(x) = 2x^3 + ax^2 + 3x - 5,`

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

`h(x) = x-2`

Now, we will find the remainders R₁ and R₂when 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)`

     ` = 2 (2)^3 + a(2)^2 + 3(2) - 5`

     `  = 16 + 4a + 6-5`

     ` = 4a + 17`

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

`R_2 = g(2)`

     ` = (2)^3 + (2)^2 - 4(2) + a`

     ` = 8 + 4 - 8 + a`

     ` = a + 4`

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

`R_1 = R_2`

`⇒ 4a + 17 = a + 4`

`⇒  4a - a = 4 -17`

`⇒  3a = - 13`

`⇒  a = -13/3`