Factorization of Polynomial Part V

Division of Algebraic Expression: Factorization Part V

Division of algebraic expressions is similar to the division of numbers. However, when we divide numbers, we use our knowledge of factors of numbers through multiplication tables but to divide algebraic expression we need to factorize their terms.

Let us consider on following algebraic expression:

1. 4a² × 3a³ = 12a⁵

=> 12a⁵ ÷ 4a² = 3a³

Also, 12a⁵ ÷ 3a³ = 4a²

2. 15m(4m + 3n) = 60m² + 45mn

=> (60m² + 45mn) ÷ (4m + 3n)

Also, (60m² + 45mn) ÷ (4m +3n) = 15m

Division of a Monomial by a Monomial

Let us consider 24a⁴ ÷ 6a

Factorizing 24a⁴ completely, we have

24a⁴ = 2 × 2 × 2 × 3 × a × a× a× a

24a⁴ ÷ 6a =2 × 2 × 2 × 3 × a × a × a × a = 2 × 2 × a × a × a = 4a³

                                    2 × 3 × a

The above division can also be done by factorizing 24a⁴ into two terms.

24a⁴ = (2 × 3 × a) × (2 × 2 × a × a × a) = 6a × 4a³

24a⁴ ÷ 6a = 6a × 4a³/6a = 4a³

Division of a Polynomial by a monomial:-

To divide a polynomial by a monomial, we divide each of its term by the monomial and add the quotient. Let us consider on the following expression:

(12a³ +15a) ÷ 3a

Factorizing 12a³ completely, we have

12a³ = 2 × 2 × 3 × a × a × a

Divide first terms by 3a, we have

 12a³ ÷ 3a = 2 × 2 ×3 × a × a × a/3 × a = 2 × 2 × a × a = 4a²

Factorizing the second term 15a, we have

15a = 3 × 5 × a

Divide the second term by 3a, we have

15a ÷ 3a = 3 × 5 × a/3 × a = 5

Therefore, combining both the quotients, we have

(12a³ + 5a) ÷ 3a = 4a² + 5

In this example, the division of algebraic expressions has been carried out by two methods.

In first method, the dividend is written as a product of factors such that the divisor is one of the factors. Then, the quotient is obtained by cancelling this common factor in both the numerators and the denominator.

In second method, each term of the dividend is divided individually by the divisor and resultant quotients of all the terms are added to get the required quotient.

Alternative Method: in this method, each term of the algebraic expression is divided individually by the divisor.

16a⁴ + 8a³ + 12a ÷ 4a = 16a⁴ + 8a³ + 12a/ 4a

                                         = 16a⁴/4a + 8a³/4a + 12a/4a

                                         = 4a³ + 2a² + 3

Division of Polynomial by a polynomial:-

There are two methods to divide a polynomial by another polynomial.

1. By factorization and 2 By long division

1. Division by factorization: This method is similar to the division of a polynomial by a monomial. To divide a polynomial by a polynomial, we factorize the dividend and the divisor (if needed) and cancel the factors common to both numerator and denominator.

Let us consider on following expression:-

(x² + 15x + 56) by (x + 8)

First we’ll factorize the dividend x² + 15x + 56

x² + 15x + 56 = x² + 8x +7x + 56

                      = x(x + 8) + 7(x +8)

                      =(x + 8)(x +7)

x² + 15x + 56 ÷  (x + 8) = x² + 15x + 56/ x + 8

                                      = (x + 8)(8 + 7)/ x + 8

                                      = x + 7

2. Division by long method:

x³ – 6x² + x + 8 by x + 1

Step1:- Write the terms of the dividend and divisor in descending order of their degree.

Step2:- Divide the first term of the dividend by the first term of the divisor. It gives us the first term of the quotient. Therefore, x³ ÷ x = x² in the first term of the quotient.

Step3:- Multiply the divisor (x + 1) by x². Write the result under proper term of the dividend and subtract the result from the dividend. To substrac, change the sign of each term in the lower row from + to – and from – to +. With the new signs add columnwise ( the new signs are written below the terms of the lower row.

We get, 1 – 7x²

Now bring down +x + 8. We get 1 – 7x² + x + 8.

This is the new dividend.

Step 4:- We now repeat step 2 to get the next term of the quotient. Dividing the first term – 7x² of the new dividend by the first term of the divisor, we get – 7x² ÷ x =   – 7x , which is the next term of the quotient.

Step 5:-Multiply the divisor (x +1) by – 7x and subtract the result from the dividend (- 7x² + x + 8) to get (8x +8), which is the next dividend.

Step 6:- Again repeat step 2 to get the next term of the quotient, i.e., 8x ÷  x = 8, which is the next term of the quotient.

Step 7:- Multiply the divisor (x +1) by 8 and subtract the result from the dividend. We get remainder 0. 

                      x² – 7x + 8____________________

x + 1               |x³ – 6x² + x + 8

                      + x³ +x²

                      _        _           

_______________________________________________

                               - 7x² + x + 8

                               - 7x² – 7x

______________   +___+___________________________     

                                               8x + 8

                                            + 8x + 8

                                             _      _

__________________________________

                                           0

___________________________________

Therefore,

(x³ – 6x² + x + 8) ÷ (x + 1) = x² – 7x + 8

Note: When we divide a number 56 by 7, we get quotient =8 and remainder =0, since 7 is a factor of 56.

Similarly, when we divide an algebraic expression by another algebraic expression and our remainder is 0, it implies that the divisor is a factor of the dividend.

In above example, we get remainder 0, therefore (x + 1) is a factor of x³ – 6x² + x + 8 and the quotient (x² – 7x + 8) is also a factor of the dividend.