Question

Using Remainder Theorem, factorise : x³ + 10x² – 37x + 26 completely.

Answer 1

By remainder theorem, 

For x = 1, the value of the given expression is the remainder. 

x³ + 10x² – 37x + 26

= (1)³ + 10(1)² – 37(1) + 26

= 1 + 10 – 37 + 26

= 37 – 37

= 0

x – 1 is a factor of x³ + 10x² – 37x + 26.

            x² + 11x – 26
${\displaystyle x-1\text{)}\overline{x^3+10x^2-37x+26}}$
           x³ – x²                           
                    11x² – 37x
                    11x² – 11x            
                             – 26x + 26
                             – 26x + 26    
                                     0

∴ x³ + 10x² – 37x + 26

= (x – 1)(x² + 11x – 26)

= (x – 1)(x² + 13x – 2x – 26)

= (x – 1)[x(x + 13) – 2(x + 13)]

= (x – 1)(x + 13)(x – 2)

∴ x³ + 10x² – 37 + 26 = (x – 1)(x + 13)(x – 2)