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