Factorizing Algebraic Expressions Lesson - Definition, Types, and Examples

Lesson Overview

• What is the Factorization of Algebraic Expressions?
• Factoring by Taking Common Factors
• Factorization of Algebraic Expressions by Regrouping Terms
• Factorization of Algebraic Expressions by Using Standard Identities
• Examples and Exercises Factorization of Algebraic Expressions

Algebraic expressions are like sentences in the language of math. They use numbers, letters (like x or y), and symbols (+, -, ×, ÷) to express relationships and patterns. 

For example,  3x + 2y  is an algebraic expression. These expressions serve as a basis for the factorization of algebra. 

Why do we factor these expressions?

Think of factorization as a way to simplify and organize. It's like separating a complex machine to understand how the different parts work together. Factorization of Algebra is helpful because:

• It can make calculations easier.

• It helps us solve equations.

• It reveals hidden patterns and relationships within the expression.

Factorization is like cleaning up in math. It helps us break down complicated expressions into smaller, more manageable parts, making them easier to work with and understand.

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What is the Factorization of Algebraic Expressions?

In mathematics, a factor is a quantity that divides another quantity exactly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because each of these numbers divides 12 evenly. A term is a single number or variable, or numbers and variables multiplied together. Terms are separated by plus or minus signs in an expression.

Factorization is the process of decomposing a mathematical object into a product of its factors. This can be applied to various objects, including numbers, polynomials, and algebraic expressions.

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Factoring by Taking Common Factors

Sometimes, all the terms in an algebraic expression have something in common, like a shared number or variable.  Factorization of algebra can simply be done by finding a common factor in an algebraic expression and pulling it out of each term, which helps in simplifying the expression. 

Let's break it down with an example:

Imagine you have the expression:  12ab + 8a

Step 1. Identify the common factor: Look at the numbers and variables in each term.

Both 12 and 8 are divisible by 4.

Both terms also have an 'a'.

So, the common factor is '4a'.

Step 2. Divide each term by the common factor:

12ab ÷ 4a = 3b

8a ÷ 4a = 2

Step 3. Rewrite the expression:

Put the common factor outside a set of parentheses: 4a(...)Inside the parentheses, write the results you got from dividing: 4a(3b + 2)

Factorization of Algebraic Expressions by Regrouping Terms

Sometimes, you can't find a common factor for all the terms in an expression. But, you might spot common factors within groups of terms. So, in this method of factorizing algebraic expressions, we rearrange (or regroup) the terms to make these common factors easier to find.

Let's illustrate with an example:

Consider the expression:  ax + by + ay + bx

Step 1. Identify potential groups:  Look for terms that seem to have something in common.

ax and bx both have an 'x'.

ay and by both have a 'y'.

Step 2. Rearrange the terms:  Group the terms with common factors:

ax + bx + ay + by

Step 3. Factor each group:

ax + bx = x(a + b)

ay + by = y(a + b)

Step 4. Look for a common factor in the new groups:

Notice that both groups now have (a + b) as a common factor!

Step 5. Factor out the common factor:

x(a + b) + y(a + b) = (a + b)(x + y)

Factorization of Algebraic Expressions by Using Standard Identities

In math, we have some special equations called "identities." These are like rules that are always true, no matter what values you use. We can use these identities as shortcuts for factorization of algebra.

Identities

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• a² - b² = (a + b)(a - b)
• (x + a)(x + b) = x² + (a + b)x + ab
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a - b)³ = a³ - 3a²b + 3ab² - b³

Let's see an example:

Factor the expression: x² + 6x + 9

Step 1. Recognize the pattern first, we can see that this expression looks like identity #1:

(a + b)² = a² + 2ab + b²

Step 2. Match the terms.

x² matches a², so a = x

9 matches b², so b = 3

6x matches 2ab, since 2 * x * 3 = 6x

Step 3. Apply the identity.

Since it fits the pattern of (a + b)², we can factor it as: (x + 3)²

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Examples and Exercises Factorization of Algebraic Expressions

1. Factorize: 15x³y² + 25x²y³

• Identify the common factor: Both terms have 5, x², and y² as common factors. The greatest common factor is 5x²y².
  
• For factorizing this algebraic expression, divide each term by the common factor:

15x³y² ÷ 5x²y² = 3x

25x²y³ ÷ 5x²y² = 5y

• Rewrite the expression: 5x²y²(3x + 5y)

2. Factorize: 2xy - 6x + 5y - 15

• Identify potential groups: (2xy - 6x) and (5y - 15)
  
• Factor each group:

2xy - 6x = 2x(y - 3)

5y - 15 = 5(y - 3)

• Factor out the common factor: 2x(y - 3) + 5(y - 3) = (y - 3)(2x + 5)

3. Factorize: 4x² - 20xy + 25y²

• Recognize the pattern: This looks like the identity: (a - b)² = a² - 2ab + b²

• Match the terms:

4x² matches a², so a = 2x

25y² matches b², so b = 5y

-20xy matches -2ab, since -2 * 2x * 5y = -20xy

• Apply the identity: (2x - 5y)²

Now, Factor the following expressions:

1. 8ab² - 12a²b
2. 3x²y + 6xy² - 9x²y²
3. p² + 8p + 16
4. m² - 49
5. 2a² + 5a + 2
6. x² + 4xy + 4y²
7. 9a² - 30ab + 25b²