Mathematics

Problem

Let

g(x)=2x^{3}+ax^{2}+bx-8

. The polynomial leaves a remainder

10

when divided by

x-2

and is exactly divisible by

x+3

. (a) Obtain two linear equations in

a

and

b

using the given information. (b) Solve these simultaneous equations to find the values of

a

and

b

. (c) Using the values of

a

and

b

, factorise

g(x)

completely. Provide complete working for each part.

Intuitive explanation

# Intuitive Explanation

The Remainder Theorem is your key tool here: when you divide a polynomial by $(x-c)$ , the remainder equals $g(c)$ . Since $g(x)$ leaves remainder 10 when divided by $x-2$ , you have $g(2)=10$ ; since it's exactly divisible by $x+3$ , you have $g(-3)=0$ . These two conditions give you two equations to solve for the unknowns $a$ and $b$ .

---

# Complete Solution

## Part (a): Obtain two linear equations in $a$ and $b$

USING THE REMAINDER THEOREM

When $g(x)$ is divided by $(x-2)$ , the remainder is $g(2)$ .

$g(2) = 2(2)^3 + a(2)^2 + b(2) - 8 = 10$ $16 + 4a + 2b - 8 = 10$ $4a + 2b + 8 = 10$ $$4a + 2b =