Question

Find ‘a’ if the two polynomials ax³ + 3x² – 9 and 2x³ + 4x + a, leaves the same remainder when divided by x + 3.

Answer 1

Find 'a' if the two polynomials ax³ + 3x² – 9 and 2x³ + 4x + a, leaves the same remainder when divided by x + 3.
The given polynomials are ax³ + 3x² – 9
and 2x³ + 4x + a
Let p(x) = ax³ + 3x² – 9
and q(x) = 2x³ + 4x + a
Given that p(x) and q(x) leave the same
remainder when divided by (x + 3),
Thus by Remainder Theorem, we have
p(–3) = q(–3)
⇒ a(–3)³ + 3(–3)² - 9 = 2(–3)³ + 4(–3) + a
⇒ –27a + 27 – 9 = –54 – 12 + a
⇒ –27a + 18 = –66 + a
⇒ –27a - a = –66 – 18
⇒ -28a = –84
⇒ a = `(84)/(28)`
∴ a = 3.