A cubic equation has the form: $ax^{3}$ + $bx^{2}$ + cx + d = 0, (a$\neq$ 0)

A cubic equation will have at least one real root it can also have three real roots but not zero roots.To factorize a cubic equation first find a factor, where in you substitute the value to see which value turns the equation to zero.

Then apply synthetic division to get a polynomial which we multiply by.

**Example:** Solve : $x^{3}$ - $5x^{2}$ -2x + 24 = 0

**Solution:** To find a factor for the given equation substitute x = - 2 we see that the equation turns out to be zero. So x = - 2 is a solution to this equation and x + 2 is a factor for the given expression.

Now $x^{3}$ - $5x^{2}$ -2x + 24 = 0 can be written in the form (x + 2) ($x^{2}$ + ax + b). Where a and b are numbers.

To find a and b we use a process called

**synthetic division.**

1. Write down the coefficients of a cubic equation in the first row of a table and draw a vertical line and write down the known root x = -2. Bring down the number 1 from the first row. 2. Multiply 1 by -2. Put the result in the second position of the blank column. Add the numbers in the second column and put in the bottom row.3. Now -7 is multiplied by -2 and the result 14 is placed in the third position of the blank row.add the numbers of the third row and multiply the result by -2.4. The process continues,We see that the final element in the bottom row is zero. The coefficients of a quadratic are 1, -7 and 12

So the quadratic is $x^{2}$ - 7x + 12

The cubic has been reduced to (x + 2) $x^{2}$ - 7x + 12

The quadratic can be factorized to give (x + 2) (x - 3) (x - 4) = 0

Therefore, the solutions are x = -2, 3 or 4.