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.

Answer 1

f(x) = 2x³ + ax² + bx – 14
∴ (x – 2) is factor of f(x)
f(2) = 0
2(2)³ + a(2)² + b(2) – 0
16 + 4a + 2b – 14 = 0
⇒ 4a + 2b = –2
2a + b = –1                 ...(i)
Also, (x – 3) it leaves remainder = 52
∴ f(3) = 52
2(3)³ + a(3)² + b(3) – 14 = 52
⇒ 54 + 9a + 3b – 14 = 52
⇒ 9a + 3b = 52 – 40
9a + 3b = 12
3a + b = 4                 ...(ii)
From (i) and (ii)
                       2a + b = –1
                       3a + b =  4
                        –      –     –    
Subtracting   –a          = –5       
∴ a = 5 put in (i)
∴ 2(5) + b = –1
⇒ b = –1 – 10
⇒ b = –11
a = 5, b = –11

⇒ `(-27a)/(8) + (27)/(4) - (3b)/(2)` = 0

⇒ –27a + 54 – 12b – 24 = 0  ...(Multiplying by 8)
⇒ –27a – 12b + 30 = 0
⇒ –27a – 12b = –30
⇒ 9a + 4b = 10     ...[Dividing by (–3)]
9a + 4b = 10           ....(i)
Again let x + 2 = 0 then x = –2
Substituting the value of x in f(x)
f(x) = ax³ + 3x² + bx – 3
f(–2) = a(–2)³ + 3(–2)² + b(–2) – 3
= –8a + 12 – 2b – 3
= –8a – 2b + 9
∵ Remainder = –3
∴ –8a – 2b + 9 = –3
⇒ –8a – 2b = –3 – 9
⇒ –8a – 2b = –12   ...(Dividing by 2)
⇒ 4a + b = 6
Multiplying (ii) by 4
                       16a + 4b = 24
                         9a + 4b = 10
Substracting    –        –      –   
                         7a           = 14

7a = 14
⇒ a = `(14)/(7)` = 2.
Substituting the value of a in (i)
9(2) + 4b = 10
⇒ 18 + 4b = 10
⇒ 4b = 10 – 18
⇒ 4b = –8

∴ b = `(-8)/(4)` = –2
Hence a = 2, b = –2
∴ f(x) = ax³ + 3x² + bx – 3
= 2x³ + 3x² – 2x – 3
∵ 2x + 3 is a factor
∴ Dividing f(x) by x + 2

`2x + 3")"overline(2x^3 + 3x^3 – 2x  – 3 )("x^2 – 1`
             2x³  + 3x²              
              –     –                             
                    –2x – 3
                    –2x – 3
                     +    +          
                         x        
∴ 2x³ + 3x² – 2x – 3   
= (2x + 3)(x² – 1) = (2x + 3)[(x²) - (1)²]
= (2x + 3)(x + 1)(x –1).