Question

While factorizing a given polynomial using the remainder and factor theorem, a student finds that (2x + 1) is a factor of 2x³ + 7x² + 2x – 3.

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

Also, factorize the given polynomial completely.

Answer 1

f(x) = 2x³ + 7x² + 2x – 3

`f(-1/2) = 2(-1/2)^3 + 7(-1/2)^2 + 2(-1/2) - 3 ≠ 0`

∴ (2x + 1) is not a factor of f(x).

`f(1/2) = 2(1/2)^3 + 7(1/2)^2 + 2(1/2) - 3 = 0`

∴ (2x – 1) is a factor of f(x)

               x² + 4x + 3
`2x - 1")"overline(2x^3 + 7x^2 + 2x - 3)`
              2x³ – x²          
                 8x² + 2x
                 8x² – 4x  
                6x – 3
                6x – 3    
                   xx
f(x) = (2x – 1)(x² + 4x + 3)

f(x) = (2x – 1)(x + 3)(x + 1)