Remainder theorem

Given that (x + 2) and (x + 3) are factors of 2x³ +ax² + 7x – b. Determine the values of “a” and “b.           (ICSE 2009)

Answer:

Let: F(x) = 2x³ +ax² + 7x – b

Since (x + 2) and (x + 3) are factors of F(x), therefore, according to Remainder theorem, we have, F (-2) = F (-3) = 0

Now, F (-2) = 2(-2)³ +a (-2)² + 7(-2) – b = 0

Or,-16 + 4a -14 – b = 0

Or, 4a –b = 30 -----------------(1)

Similarly, F (-3) = 2(-3)³ +a (-3)² + 7(-3) – b = 0

Or, F (-3) = -54 +9a – 21 – b = 0

Or, 9a – b = 75 ------------------ (2)

Solving equation (1) and (2), we get

-5a = -45

Or a = 9

Therefore, b = 9x9 – 33 =48.

Answers: a = 9, b = 48