The sums of factorization || Examples of factorization

Factorization is an essential topic of Algebra and business mathematics, and it is therefore referred to as Algebra factorization too. Factorization is executed by various methods. Today we learn the following factorization with examples. 

• Factorization by a common term

• Factorization by making pairs

• Factorization by any suitable formula

• Factorization by breaking the middle term

• Factorization by completing the square method

• Factorization by Quadratic formula.

What is factorization by a common term?

Factorization by the common term is performed by taking out any common variable or number in an equation. The common term is an important part of the factorization by a common term. In this factorization, there must be a variable like x, y, z, or any number that can be common.

Examples of factorization by common term:

• 6𝑥²-24y+48

→6(𝑥²-4y+8) Ans

• 4𝑥²y+5𝑥y²-3𝑥y 

→𝑥y(4𝑥+5y-3) Ans

What is Factorization by making pairs?

In Algebra factorization by making pairs is to make pairs or divide the terms into two pairs. In factorization by making pairs, there are usually four terms we make the pair accordingly to get the like and common terms.

Examples of factorization by making pairs:

• 16𝑥²+32𝑥+3𝑥+6

→(16𝑥²+32𝑥)+(3𝑥+6)

→16𝑥(𝑥+2)+3(𝑥+2)

→(16𝑥+3)(𝑥+2) Ans

• 11y²+33y+7y+21

→(11y²+33y)+(7y+21)

→11y(y+3)+7(y+3) Ans

 

What is factorization by any suitable formula?

In mathematics, factorization by any suitable formula is solved by using an appropriate formula. Factorization by any suitable formula can also be executed differently, but applying the formula will be time-saving for the students.

Examples of factorization by any suitable formula:

• 169𝑥²-156𝑥+36

∴ formula=  (𝑥-y)²= 𝑥²-2𝑥y+y²

→(13𝑥)²-2(13𝑥)(6)+(6)²

→(13𝑥-6)² Ans

• 324𝑥²-49

∴ formula=  (𝑥)²-(y)² =  (𝑥-y)(𝑥+y)

→(18𝑥)²-(7)² = (18𝑥-7)(18𝑥+7)

→(18𝑥-7)(18𝑥+7) Ans

What is the factorization by breaking the middle term?

Factorization by breaking the middle term is quite similar to factorization by making pair. In factorization by breaking the middle term, we break the middle term into two parts.

How to break the middle in factorization?

It is very simple and easy to break the middle term in factorization. Multiply the first term with the last term of the given equation. Then, make the factors of that product that give the middle term number if we add both factors. Let's clear with an example if the given equation is 7y²+16y+4. Now multiply 7 by 4, and the answer will be 28. After that, get the factors of 28 i.e., 2,4,7,14, 28.

If we multiply 2 by 14 we will get 28 and in the same way, if we add 2 and 14 we will get 2+14= 16. Thus, 2y+14y will replace 16y, and we will write the above equation in this way 7y²+2y+14y+4.

Examples of factorization by breaking the middle term:

• 8𝑥²+8𝑥+2

→ 8𝑥²+4𝑥+4𝑥+2

→ (8𝑥²+4𝑥)+(4𝑥+2)

→ 4𝑥(2𝑥+2)+2(2𝑥+2)

→ (4𝑥+2)(2𝑥+2) Ans

 

• 10y²+57𝑥y+11𝑥²

→ 10y²+2𝑥y+55𝑥y+11𝑥²

→ (10y²+2𝑥y)+(55𝑥y+11𝑥²)

→ 2y(5y+𝑥)+11𝑥(5y+𝑥)

→ (2y+11𝑥)(5y+𝑥) Ans

What is the factorization by a completing square method?

Factorization by a completing square method is quite different from other factorizations. There are various steps to completing square factorization.

Simple steps of a completing square factorization:

1. co-efficient of 𝑥² should be 1.

it should be like this 𝑥² if it is not then we will divide the coefficient of 𝑥² with an entire equation. For instance, if it is 5𝑥² then will divide 5 by the given equation.

2. Constant term will be moved to = side

3. coefficient of 𝑥 will be multiple with 0.5 or ½ and then it will be squared the final answer will be added to both sides.

Examples of factorization by a completing square method?

• 10𝑥²+80𝑥-120=0

→Divide 10 

→10/10𝑥²+80/10𝑥-120/10=0

→𝑥²+8𝑥-12=0

→ move -12

→𝑥²+8𝑥=12

→Take the co-efficient of 𝑥 i.e., 8, and multiply it with 1/2

→[8x1/2]2 

→[4]2

→16

→add 16 on both sides

→𝑥²+8𝑥+16=12+16

→𝑥²+8𝑥+16=28

→(𝑥+4)²=28

→Take square root on b/s

→√(𝑥+4)²=√28

→𝑥+4= ±2√7

→𝑥= ±2√7-4

→𝑥= ±2(√7-2)  Ans

• 𝑥²+24𝑥-25=0

→ move -25

→𝑥²+24𝑥=25

→Take the co-efficient of 𝑥 i.e., 24, and multiply it with 1/2

→[24x1/2]2 

→[12]2

→144

→add 144 on both sides

→𝑥²+24𝑥+144=25+144

→𝑥²+24𝑥+144=169

→(𝑥+12)²=169

→Take square root on b/s

→√(𝑥+12)²=√169

→𝑥+12= ±13

→𝑥= ±13-12

→𝑥= +13-12 | -13-12

→𝑥= +1 | -25

→S.S= {1,-25} Ans

What is the factorization by the Quadratic formula?

The factorization by Quadratic formula is executed by using a formula given below.

We can apply the quadratic formula if an equation is in the form of 𝑥²+4𝑥-49=0

Examples of factorization by Quadratic formula:

• 𝑥²+15𝑥-9=0

• 𝑥²+2𝑥-48=0