Question

If the polynomials $a{x}^3+3x^2$ - 13 and $2x^3$ - 5x + a, when divided by (x-2), leave the same remainders, find the value of a.

Answer 1

Let p(x) = $a{x}^3+3x^2$ - 13

q(x) = $2x^3$ - 5x + a

and divisor g(x) = x - 2

x - 2 = 0 $\Rightarrow$ x = 2

$\therefore$ Remainder = p(2) = $a(2{)}^3+3(2{)}^2$ -13

= 8a + 12 -13 = 8a -1

and q(2) = $2(2{)}^3-5\times$ 2 + a = 16 - 10 + a = 6 + a

$\because$ In each case remainder is same

$\therefore$ 8a - 1 = 6 + a

8a - a = 6 + 1

$\Rightarrow 7a=7\Rightarrow a=\frac{7}{7}=1$

$\therefore$ a = 1