Question

When the polynomial x³ + 2x² – 5ax – 7 is divided by (x – 1), the remainder is A and when the polynomial x³ + ax² – 12x + 16 is divided by (x + 2), the remainder is B. Find the value of ‘a’ if 2A + B = 0.

Answer 1

It is given that when the polynomial x³ + 2x² – 5ax – 7 is divided by (x – 1), the remainder is A.

∴ (1)³ + 2(1)² – 5a(1) – 7 = A

1 + 2 – 5a – 7 = A

–5a – 4 = A   ...(i)

It is also given that when the polynomial x³ + ax² – 12x + 16 is divided by (x + 2), the remainder is B.

∴ x³ + ax² – 12x + 16 = B

(–2)³ + a(–2)² – 12(–2) + 16 = B

–8 + 4a + 24 + 16 = B

4a + 32 = B  ...(ii)

It is also given that 2A + B = 0

Using (i) and (ii), we get,

2(–5a – 4) + 4a + 32 = 0

–10a – 8 + 4a + 32 = 0

–6a + 24 = 0

6a = 24

a = 4