In this section, we shall factorize polynomials of degree 3. This is often one by dividing a polynomial by one of its factors, then making use of the fact that: P(x) ÷ h(x) = g(x) →px=gx×hx

Example

Given that, the polynomial x – 1 is a factor of P(x).

 P(x) = 2x³-3x²+4kx-3. Find the value of a and hence factorize P(x) completely.

Solution

X + 1 = 0

X = -1

P(x) = 0, since x is a factor

P(-1) = 2(-1)³-3(-1)²+4k(-1)-3=0

           = -2-3-4k-3= 0

→4k=-8, k=-2

P(x) = 2x³ – 3x² + 4(-2)x – 3

 P(x) = 2x³ – 3x² - 8x – 3

Factorize;

             2x²-5x-3

X + 1/2x³ – 3x² - 8x – 3

• (2x³ + 2x²)    

            -5x² – 8x

•  (-5x² – 5x)

           -3x – 3

•       (-3x – 3)

        _        _

→px=x+12x2-5x-3

But factorizing, 2x²-5x-3

                           (2x²-6x) + (x-3)

                         2x(x – 3) +1(x – 3)

                      (x – 3) (2x + 1)

2x² – 5x – 3 = (x – 3) (2x + 1)

Therefore; p(X) = (x +1) (x – 3) (2x + 1)