Factorization of Algebraic Expression

In algebra, factorization is a fundamental concept that helps in simplifying expressions and solving equations. Factorization involves breaking down algebraic expressions into simpler components, which aids in understanding their structure and properties.

In this article, we'll look at basic methods and examples for factorizing algebraic expressions.

What are Expressions?

An algeabric expression connects variables and constants by algebraic operations of addition, subtraction, multiplication, and division. For example: x + 2y; 4x - y +5.

What is Factorization of Algebraic Expressions?

A number can be expressed as the product of any two numbers using the term "factor." If we have to find the factors for any mathematical object an integer, a polynomial, or an algebraic expression are all done under the process of factorization. Therefore, determining the factors of a particular algebraic expression is referred to as factorization of the algebraic expression.

For Example: Lets' factorize the expression; 3x² + 9x

First step in factorizing this expression is to find common factors between the terms. Here, the common factor between the two terms is 3x

So, we will separate the 3x from both the terms

3x² + 9x = 3x(x + 9)

Therefore, The expression is factorized in two terms 3x and x + 9

Terminology Related to Factorization

Some of the important keywords related to the Factorization are:

• Factor: A number or expression's factor is the value that divides it equally without producing leftovers. For Example: 2 and 3 are factors of 6.
• Prime Factorization: Prime factorization is the method that allows every integer that is greater than 1 to be expressed as the product of prime integers.
• Greatest common factor: It stands for the greatest number that can be used to divide each statement or integer without producing a residue.
• Comparison of Squares: An algebraic statement of the form a² - b² can be factored into the product of two binomials: (a - b)(a + b).

How to Factorize Algebraic Expressions?

There are several techniques employed in factorization, each tailored to different types of expressions and scenarios. Some of these techniques are:

• Factorization By Common Factors
• Factorization By Regrouping Terms
• Factorization By Standard Identities

Factorization using Common Factors

One of the simplest methods involves identifying and factoring out the greatest common factor (GCF) from an expression. This involves finding the largest term that divides evenly into each term of the expression.

For Example: -2y² + 16y

So, algebraic expression can be written as: -2 . y . y + 2 . 8 . y

After taking 2y common from both sides, we get 2y( -y + 8)

Thus, -2y² + 16y = 2y( -y + 8)

Alegabric expression -2y² + 16y after expanding is 2y( -y + 8).

Therefore, we can factorize using common factors.

Factoring by Regrouping Terms

There may not be a common factor for every term of the algebraic expressions.

For Example: So, lets take the algebraic statement 14a + c -ca – 14.

Here, first and last terms in this expression share the factor "14".

However other terms do not share a specific factors.

Similarly, common factor between second and third terms is c.

As a result, terms can be rearranged as:

14a + c – ca – 14

= 14a – 14 – ac + c

= 14(a -1) – c(a -1) {Here, (a - 1) is a common term}

After regrouping it can be written as: (14 - c) (a - 1)

Thus, 14a + c -ca – 14 = (14 - c) (a - 1)

Therefore, by regrouping terms we can factorize algebraic Expression

Factorization Using Standard Identities

In mathematics, an identity is an equality relation that is valid for any value of a variable. So, let's talk about the important identities;

• Difference of Squares: a² - b² = (a+b)(a−b)

• Perfect Square Trinomial: a² + 2ab + b² = (a + b)²

• Perfect Square Trinomial: a² - 2ab + b² = (a + b)²

• Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)

• Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)

When any value for a and b is substituted, the provided equation's two sides stay the same. These equations are hence referred to as identities.

For Example: 9x² + 3m² + 12mx

To factorize 9x² + 3m² + 12mx, we start by determining the common factor 3 in this expression. Then the expression is rearranged to see if it still follows the structure of a perfect square trinomial or not.

27x² + 3m² + 18mx = 3(9x² + m² + 6mx)

Recognize that 9x² + m² + 6mx resembles the pattern of (a + b)² = a² + 2ab + b²

Here, a= 3x and b = m.

3(9x² + m² + 6mx) = 3[(3x)² + m² + 2. 3x . m] 3(3x + m)²

Therefore, the factorization of 9x² + 3m² + 12mx is 3(3x + m)² .

Real-Life Examples of Factorization

Factorization finds practical applications in diverse fields, illustrating its relevance beyond theoretical mathematics.

• Application in Finance: In finance, factorization techniques are used to analyze investment portfolios, assess risk, and model financial instruments.

• Application in Physics: In physics, factorization is utilized to simplify equations describing physical phenomena, aiding in the derivation of fundamental principles and laws.

Read More: Real-Life Applications of Prime Factorization

Key Points on Factorization of Algebraic Expression

• Factorization simplifies algebraic expressions, making them easier to work with and understand.

• It's essential for solving equations and uncovering patterns in mathematics.

• Factorization has practical uses in various fields, from finance to physics, making it a valuable skill beyond the classroom.

Conclusion

In conclusion, understanding the process of factorization in algebraic expressions is crucial for simplifying complex equations and revealing underlying patterns and relationships. By factoring expressions, we can efficiently solve equations, identify common factors, and manipulate expressions to better understand their behavior.

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Examples on Factorization of Algebraic Expression

Example 1: x² + 5x + 6

Solution:

To factorize the expression x² + 5x + 6, we need to find two numbers that multiply to give us 6 and add to give us 5.

The two numbers are 2 and 3, because 2 × 3 = 6 and 2 + 3 = 5.

So, we can rewrite the expression as:

x² + 5x + 6 = (x + 2)(x + 3)

Thus, the factorization of x² + 5x + 6 is (x + 2)(x + 3).

Example 2: x² - 4

Solution:

To factorize the expression x² - 4, we can use identity i.e.,

a² - b² = (a+b)(a−b)

So, x² - 4 = x² - 2² = (x + 2)(x - 2).

Example 3: x³ + 3x² + 2x + 6

Solution:

To factorize the expression x³ + 3x² + 2x + 6, we group the terms:

(x³ + 3x²) + (2x + 6)

= x²(x + 3) + 2(x + 3)

= (x² + 2)(x + 3)

So, x³ + 3x² + 2x + 6 = (x² + 2)(x + 3).