## SOLVING POLYNOMIAL EQUATIONS BY FACTORING

In this page solving polynomial equations by factoring we are going to see how to factor a polynomial degree 3.

## Procedure of solving polynomial equations by factoring:

Step 1:  Arrange the dividend and the divisor according to the descending powers of x and then write the coefficients of dividend in the first zero. Insert 0 for missing terms.

Step 2:  Find out the zero of the divisor.

Step 3:  Put 0 for the first entry in the second row.

Step 4:  Write down the quotient and remainder accordingly. All the entries except the last one  in the third row constitute the coefficients of the quotient.

## Example of solving polynomial equations by factoring.

Question 1

Factorize each of the following polynomial x³ - 2 x² - 5 x + 6

Solution

Let p (x) = x³ - 2 x² - 5 x + 6

x = 1

p (1) = 1³ - 2 (1)² - 5 (1) + 6

= 1 - 2 - 5 + 6

= 7 - 7

= 0

So we can decide (x - 1) is a factor. To find other two factors we have to use synthetic division.

So the factors are (x - 1) and (x² - x - 6). By factoring this quadratic equation we get  (x - 3) (x + 2)

Therefore the required three factors are (x - 1) (x - 3) (x + 2)

Question 2

Factorize each of the following polynomial 4 x³ - 7 x + 3

Solution

Let p (x) = 4 x³ - 7 x + 3

x = 1

p (1) = 4 (1)³ -7 (1) + 3

= 4 - 7 + 3

= 7 - 7

= 0

So we can decide (x - 1) is a factor. To find other two factors we have to use synthetic division.

So the factors are (x - 1) and (4 x² - 4 x - 3). By factoring this quadratic equation we get  (2 x + 3) (2 x - 1)

Therefore the required three factors are (x - 1) (2 x + 3) (2 x - 1)

Question 3

Factorize each of the following polynomial  x³ - 23 x² + 142 x - 120

Solution

Let p (x) =  x³ - 23 x² + 142 x - 120

x = 1

p (1) = 1³ - 23 (1)² + 142 (1) - 120

= 1 - 23 + 142 - 120

= 143 - 143

= 0

So we can decide (x - 1) is a factor. To find other two factors we have to use synthetic division.

So the factors are (x - 1) and (x² - 22 x + 120). By factoring this quadratic equation we get  (x - 12) (x - 10)

Therefore the required three factors are (x - 1) (x - 12) (x - 10)