Question

for what value of a the polynomial 4x^3-ax^2+2x-1 and 3x^3+7x^2 -8x+a leave the same
remainder when divided by x-1​

Answer 1

$\large\underline{\sf{Solution-}}$

Given polynomials are

$\rm :\longmapsto\: {4x}^{3} - {ax}^{2} + 2x - 1$

and

$\rm :\longmapsto\: {3x}^{3} + {7x}^{2} - 8x + a$

Let suppose that,

$\rm :\longmapsto\: p(x) = {4x}^{3} - {ax}^{2} + 2x - 1$

and

$\rm :\longmapsto\: q(x) = {3x}^{3} + {7x}^{2} - 8x + a$

Now, it is further given that when p(x) and q(x) are divided by (x-1), they leave the same remainder.

We know, By Remainder Theorem,

When a polynomial f(x) is divided by linear polynomial (x - a), the remainder is f(a).

So, using this remainder theorem,

$\rm :\longmapsto\:p(1) = q(1)$

$\rm :\longmapsto\: {4} - {a} + 2 - 1 = 3 + 7 - 8 + a$

$\rm :\longmapsto\:5 - a = 2 + a$

$\rm :\longmapsto\: - 2a = - 3$

$\rm :\longmapsto\: 2a = 3$

$\rm :\longmapsto\:a = \dfrac{3}{2}$