If the polynomial
Question:

If the polynomial $2 x^{3}+a x^{2}+3 x-5$ and $x^{3}+x^{2}-4 x+a$ leave the same remainder when divided by $x-2$, Find the value of a

Solution:

Given, the polymials are

$f(x)=2 x^{3}+a x^{2}+3 x-5$

$p(x)=x^{3}+x^{2}-4 x+a$

The remainders are f(2) and p(2) when f(x) and p(x) are divided by x – 2

We know that,

f(2) = p(2)  (given in problem)

we need to calculate f(2) and p(2)

for, f(2)

substitute (x = 2) in f(x)

$f(2)=2(2)^{3}+a(2)^{2}+3(2)-5$

= (2 * 8) + 4a + 6 – 5

= 16 + 4a + 1

= 4a + 17 …. 1

for, p(2)

substitute (x = 2) in p(x)

$p(2)=2^{3}+2^{2}-4(2)+a$

= 8 + 4 – 8 + a

= 4 + a …. 2

Since, f(2) = p(2)

Equate eqn 1 and 2

⟹ 4a + 17 = 4 + a

⟹ 4a – a = 4 – 17

⟹ 3a = -13

⟹ a = -13/3

The value of a = −13/3