Factorizing Algebraic Expressions

What is Factorizing?

Factorizing is the process of expressing an algebraic expression as a product of its factors. This is the reverse of expanding an expression, where you take a complex expression and rewrite it as a product of simpler expressions.

Steps to Factorize an Expression

Find the greatest common factor (GCF) of all the terms in the expression.
Factor out the GCF from each term.
Write the expression as a product of the GCF and the simplified expression in parentheses.

Examples

Example 1:

Factorize the expression: \( 6x + 12 \)

Solution:

Step 1: Find the GCF of 6x and 12. The GCF is 6.

Step 2: Factor out the 6 from each term.

\( 6x + 12 = 6(x + 2) \)

Step 3: The factorized expression is \( 6(x + 2) \).

Example 2:

Factorize the expression: \( 8y^2 - 4y \)

Solution:

Step 1: Find the GCF of 8y² and -4y. The GCF is 4y.

Step 2: Factor out the 4y from each term.

\( 8y^2 - 4y = 4y(2y - 1) \)

Step 3: The factorized expression is \( 4y(2y - 1) \).

Example 3:

Factorize the expression: \( 15a^2b - 10ab^2 \)

Solution:

Step 1: Find the GCF of 15a²b and -10ab². The GCF is 5ab.

Step 2: Factor out the 5ab from each term.

\( 15a^2b - 10ab^2 = 5ab(3a - 2b) \)

Step 3: The factorized expression is \( 5ab(3a - 2b) \).

Exercises

Exercise 1:

Factorize the expression: \( 9x + 18 \)

Exercise 2:

Factorize the expression: \( 12y^2 - 8y \)

Exercise 3:

Factorize the expression: \( 20a^2b - 15ab^2 \)

Solutions to Exercises

Solution 1:

Factorized Expression: \( 9(x + 2) \)

Solution 2:

Factorized Expression: \( 4y(3y - 2) \)

Solution 3:

Factorized Expression: \( 5ab(4a - 3b) \)

Lesson Summary

In this lesson, we learned how to factorize algebraic expressions by identifying the greatest common factor (GCF) and rewriting the expression as a product of the GCF and the simplified expression inside parentheses. Factorizing helps simplify expressions and solve equations more easily.