Question

The polynomial 3x³ + 8x² – 15x + k has (x – 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

Answer 1

Given, P(x) = 3x³ + 8x² – 15x + k

Put x – 1 = 0

x = 1

Now, P(1) = 3(1)³ + 8(1)² – 15(1) + k = 0

${\displaystyle \Rightarrow }$ 3 + 8 – 15 + k = 0

${\displaystyle \Rightarrow }$ – 4 + k = 0

${\displaystyle \Rightarrow }$ k = 4

Hence, k = 4

Factorization:

            P(x) = 3x³ + 8x² – 15x + 4
${\displaystyle x-1\text{)}\overline{3x^3+8x^2-15x+4}(3x^2+11x-4}$
           3x³ – 3x²
          –      +                  
                 11x² – 15x
                 11x² – 11x
          –      +                  
               – 4x + 4
               – 4x + 4
                   +  –            

∴ 3x³ + 8x² – 15x + 4 = (x – 1)(3x² + 11x – 4)

= (x – 1)(3x² + 12x – x – 4)

= (x – 1)[3x(x + 4) – 1(x + 4)]

= (x – 1)(3x – 1)(x + 4)