Question:
If the polynomials $a x^{3}+3 x^{2}-13$ and $2 x^{3}-5 x+a$ when divided by $(x-2)$ leave the same remainder, Find the value of a
Solution:
Here, the polynomials are
$f(x)=a x^{3}+3 x^{2}-13$
$p(x)=2 x^{3}-5 x+a$
equate, x - 2 = 0
x = 2
substitute the value of x in f(x) and p(x)
$f(2)=(2)^{3}+3(2)^{2}-13$
= 8a + 12 - 13
= 8a - 1 ..... 1
$p(2)=2(2)^{3}-5(2)+a$
= 16 - 10 + a
= 6 + a .... 2
f(2) = p(2)
⟹ 8a - 1 = 6 + a
⟹ 8a - a = 6 + 1
⟹ 7a = 7
⟹ a = 1
The value of a = 1