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)