Question

Given that x – 2 and x + 1 are factors of f(x) = x³ + 3x² + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).

Answer 1

f(x)= x³ + 3x² + ax + b

Since, (x – 2) is a factor of f(x), f(2) = 0 

`\implies` (2)³ + 3(2)² + a(2) + b = 0 

`\implies` 8 + 12 + 2a + b = 0 

`\implies` 2a + b + 20 = 0  ...(i) 

Since, (x + 1) is a factor of f(x), f(–1) = 0 

`\implies` (–1)³ + 3(–1)² + a(–1) + b = 0 

`\implies` –1 + 3 – a + b = 0

`\implies` –a + b + 2 = 0   ...(ii) 

Subtracting (ii) from (i), we get, 

3a + 18 = 0 

 ⇒ a = – 6 

Substituting the value of a in (ii), we get, 

b = a – 2

= – 6 – 2

= – 8  

∴ f(x) = x³ + 3x² – 6x – 8 

Now, for x = –1, 

f(x) = f(–1)

= (–1)³ + 3(–1)² – 6(–1) – 8

= –1 + 3 + 6 – 8

= 0 

Hence, (x + 1) is a factor of f(x).

            x² + 2x – 8
`x + 1")"overline(x^3 + 3x^2 - 6x - 8)`
          x³ + x²                     
                 2x² – 6x
                 2x² + 2x            
                        – 8x – 8      
                        – 8x – 8      
                               0         

∴ x³ + 3x² – 6x – 8 = (x + 1)(x² + 2x – 8) 

= (x + 1)(x² + 4x – 2x – 8)

= (x + 1)[x(x + 4) – 2(x + 4)] 

= (x + 1)(x + 4)(x – 2)