**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.

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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)