Question

While factorizing a given polynomial using the remainder and factor theorem, a student finds that (x + 3) is a factor of 2x³ – x² – 5x – 2.

1. Is the student’s solution correct in stating that (x + 3) is a factor of the given polynomial?
2. Give a valid reason for your answer.
3. factorize the given polynomial completely.

Answer 1

Given polynomial: 2x³ – x² – 5x – 2.

By factor theorem,

If x – a is the factor of polynomial f(x), then remainder f(a) = 0.

If (x + 3) is a factor, then by Factor Theorem :

f(−3) = 0.

Substituting x = -3 in polynomial we get, remainder = 0 :

⇒ 2(–3)³ – (–3)² – 5(–3) – 2

⇒ 2(–27) – 9 + 15 – 2

⇒ –54 – 9 + 15 – 2

⇒ –50.

Since f(–3) ≠ 0, Remainder ≠ 0.

Hence, the student's solution is incorrect.

Factorizing,

Substituting x = 2 in polynomial we get,

f(2) = 2(2)³ – (2)² – 5(2) – 2

= 16 – 4 – 10 – 2

= 16 – 16

= 0.

Since f(2) = 0, (x - 2) is a factor 2x³ – x² – 5x – 2.

On dividing, 2x³ – x² – 5x – 2 by x – 2,

            2x² + 3x + 1
`x  –  2 ")"overline(2x^3  –  x^2  –  5x  –  2)`
          2x³ – 4x² 
        –       +                   
                   3x² – 5x
                   3x² – 6x       
                –       +             
                              x – 2      
                              x – 2      
                          –     +       
                                   0      

2x³ – x² – 5x – 2 = (x – 2)(2x² + 3x + 1)

= (x – 2)(2x² + 2x + x + 1)

= (x – 2)[2x(x + 1) + 1(x + 1)]

= (x – 2)(x + 1)(2x + 1).

Hence, 2x³ – x² – 5x – 2 = (x – 2)(x + 1)(2x + 1).