Question

The expression 4x³ – bx² + x – c leaves remainders 0 and 30 when divided by x + 1 and 2x – 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely. 

Answer 1

Let f(x) = 4x³ – bx² + x – c 

It is given that when f(x) is divided by (x + 1), the remainder is 0.

∴ f(–1) = 0 

4(–1)³ – b(–1)² + (–1) – c = 0 

– 4 – b – 1 – c = 0 

b + c + 5 = 0  ...(i) 

It is given that when f(x) is divided by (2x – 3), the remainder is 30. 

∴ `f(3/2) = 30 ` 

`4(3/2)^3 - b(3/2)^2 + (3/2) - c = 30` 

`27/2 - (9b)/4 + 3/2 - c = 30`

54 – 9b + 6 – 4c – 120 = 0

9b + 4c + 60 = 0   ...(ii)  

Multiplying (i) by 4 and subtracting it from (ii), we get, 

5b + 40 = 0

b = – 8

Substituting the value of b in (i), we get, 

c = –5 + 8 = 3 

Therefore, Let f(x) = 4x³ – 8x² + x – 3 

Now, for x = –1, we get, 

f(x) = f(–1)

= 4(–1)³ + 8(–1)² + (–1) – 3

= – 4 + 8 – 1 – 3

= 0 

Hence, (x + 1) is a factor of f(x). 

            4x² + 4x – 3
`x + 1")"overline(4x^3 + 8x^2 + x - 3)`
           4x³ + 4x²                
                     4x² + x
                     4x² + 4x        
                            – 3x – 3
                            – 3x – 3   
                                  0

∴ 4x³ + 8x² + x – 3 = (x + 1)(4x² + 4x – 3)

= (x + 1)(4x² + 6x – 2x – 3)

= (x + 1)[2x(2x + 3) – (2x + 3)]

= (x + 1)(2x + 3)(2x – 1)