Question

The polynomials $a{x}^3+3x^2-3$ and $2x^3-5x+a,$ when divided by x - 4 leave the same remainder in each case. Find the value of a.

Answer 1

Given polynomials

P($x_1$ )= a$x^3$ +3$x^2$ - 3 and
p($x_2$ )= 2$x^3$ - 5x + a

It is also given that these two polynomials leave the same remainder when divided by (x - 4).

i.e., (x-4) is the zero of the polynomial so, x=4

Now put the value of 'x' in the polynomials,

As both the Eq. have the same remainder so,

p($x_1$ )=p($x_2$ )

$\Rightarrow$ a($4^3$ ) + 3($4^2$ ) - 3 = 2($4^3$ ) - 5(4) + a

64a + 48-3 = 128 - 20 + a

64a - a = 108 - 45

63 a = 63

a = 1