Question

If (x – 2) is a factor of the expression 2x³ + ax² + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.

If (x – 2) is a factor of 2x³ + ax² + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.

Answer 1

Since (x – 2) is a factor of polynomial 2x³ + ax² + bx – 14, we have 

2(2)³ + a(2)² + b(2) – 14 = 0

⇒ 16 + 4a + 2b – 14 = 0

⇒ 4a + 2b + 2 = 0

Dividing the entire equation by 2,

⇒ 2a + b = –1   ...(1)

On dividing by (x – 3), the polynomial 2x³ + ax² + bx – 14 leaves remainder 52, 

2(3)³ + a(3)² + b(3) – 14 = 52

⇒ 54 + 9a + 3b – 14 = 52

⇒ 9a + 3b = 52 – 40

⇒ 9a + 3b = 12

Dividing the entire equation by 3,

⇒ 3a + b = 4   ...(2) 

Subtracting (1) and (2), we get 

2a + b = –1
3a + b = 4
 –   –     –     
–a      = –5   

Substituting a = 5 in (1), we get 

2 × 5 + b = –1

⇒ 10 + b = –1

⇒ b = –11 

Hence, a = 5 and b = –11.