Question

If a polynomial x³ + 2x² – ax + b leaves a remainder –6 when divided by x + 1 and the same polynomial has x – 2 as a factor, then find the values of a and b.

Answer 1

Given: f(x) = x³ + 2x² – ax + b. When divided by x + 1 the remainder is –6 and x – 2 is a factor, so the remainder is 0 at x = 2.

Step wise calculation:

1. From divisor x + 1 → x = –1:

f(–1) = (–1)³ + 2(–1)² – a(–1) + b 

= –1 + 2 + a + b 

= a + b + 1

Set equal to –6: 

a + b + 1 = –6 

⇒ a + b = –7

2. From factor x – 2 → x = 2:

f(2) = 2³ + 2 × 2² – a × 2 + b 

= 8 + 8 – 2a + b

= 16 – 2a + b

Set equal to 0:

16 – 2a + b = 0 

⇒ –2a + b = –16

3. Solve the system:

From –2a + b = –16 

⇒ b = 2a – 16

Substitute into a + b = –7: 

a + (2a – 16) = –7 

⇒ 3a – 16 = –7 

⇒ 3a = 9

⇒ a = 3

Then b = 2(3) – 16

= 6 – 16 

= –10