Question

find the polynomial x^3+ax^2+5 and x^3-2x^2+a are divided by (x+2) leave the same remainder,find the value of a.​

Answer 1

Idea

• Remainder Theorem

P(x) leaves the remainder P(α) after division by (x-α).

P(α) means x=α is substituted in P(x).

Solving

After division by (x+2)

• $\sf{x^3+ax^2+5}$ leaves $\sf{-8+4a+5}$
• $\sf{x^3-2x^2+a}$ leaves $\sf{-8-8+a}$

The two remainders are the following

• $\sf{4a-3}$ ...(1)
• $\sf{a-16}$ ...(2)

From the question, we know that (1) and (2) are equal.

We establish an equation

$\sf{4a-3=a-16}$

∴$\sf{a=-\dfrac{13}{3} }$