Factorization is a fundamental skill in algebra that allows us to break down complex expressions into simpler components. At HappyMath, we believe that mastering factorization techniques is essential for success in algebra and higher mathematics. Whether you're solving equations, simplifying expressions, or analyzing functions, factorization serves as a powerful tool in your mathematical toolkit.
In this comprehensive guide, we'll explore various factorization techniques with clear examples and step-by-step solutions.
Taking Out Common Factors: The Foundation of Factorization
The most basic factorization technique involves identifying and taking out common factors from all terms in an expression.
Step-by-Step Approach:
Identify the common factor in all terms
Factor out this common term
Write the expression as the common factor multiplied by the remaining terms
Examples:
Example 1: x² - 4x - 12
Identify the common factor: There's no common factor for all terms
This requires a different factorization approach (we'll cover this later)
Example 2: 4x - 12
Identify the common factor: 4 is common to both terms
Factor out: 4(x - 3)
Note: When factoring out a negative, the signs inside the bracket change!
Example 3: -9x - 36
Identify the common factor: -9 is common to both terms
Factor out: -9(x + 4)
Be careful with signs: When -9 is factored out, -36 becomes +4 inside the bracket
Example 4: 10a² + 4a
Identify common factors: 2 and a appear in both terms
Factor out: 2a(5a + 2)
Example 5: 17a² + 34a
Identify common factors: 17a appears in both terms
Factor out: 17a(a + 2)
Factoring More Complex Expressions
When expressions have multiple terms, we must carefully identify common factors across all terms.
Example 6: 3x - 18
Identify common factor: 3 appears in both terms
Factor out: 3(x - 6)
Example 7: 4x + 20
Identify common factor: 4 appears in both terms
Factor out: 4(x + 5)
Example 8: 7ab + 7b
Identify common factor: 7b appears in both terms
Factor out: 7b(a + 1)
Example 9: 9a - 15
Identify common factor: 3 appears in both terms
Factor out: 3(3a - 5)
Example 10: -5x - 30
Identify common factor: -5 appears in both terms
Factor out: -5(x + 6)
Remember: When factoring out a negative, signs inside the bracket change!
Handling Expressions with Negative Common Factors
Factoring out negative common factors requires special attention to signs.
Example 11: -4y - 2
Identify common factor: -2 appears in both terms
Factor out: -2(2y + 1)
Notice how -4y becomes 2y and -2 becomes 1 inside the bracket
Example 12: -3x + 12
Identify common factor: -3 appears in both terms
Factor out: -3(x - 4)
The minus sign changes both terms inside the bracket
Factoring Expressions with Multiple Variables
When expressions contain multiple variables, we need to identify all common variables and constants.
Example 13: 12a - 3
Identify common factor: 3 appears in both terms
Factor out: 3(4a - 1)
Example 14: -by - 2ab
Identify common factor: -b appears in both terms
Factor out: -b(y + 2a)
Example 15: xy - 2x
Identify common factor: x appears in both terms
Factor out: x(y - 2)
Example 16: 6b² - 18b
Identify common factors: 6b appears in both terms
Factor out: 6b(b - 3)
Example 17: -7a² + 21a
Identify common factors: 7a appears in both terms
Factor out: 7a(-a + 3) or -7a(a - 3)
Both forms are correct; it's just a matter of preference
Factoring Expressions with Multiple Terms
When dealing with expressions having three or more terms, we still look for common factors across all terms.
Example 18: 5a² - 5a
Identify common factors: 5a appears in both terms
Factor out: 5a(a - 1)
Example 19: 12x² + 30x
Identify common factors: 6x appears in both terms
Factor out: 6x(2x + 5)
Example 20: -2x² - 2x
Identify common factors: -2x appears in both terms
Factor out: -2x(x + 1)
Example 21: -4y² - 8y
Identify common factors: -4y appears in both terms
Factor out: -4y(y + 2)
Factoring with Multiple Variable Terms
Expressions with multiple variables require careful identification of all common terms.
Example 22: ab² - a²b
Identify common factors: ab appears in both terms
Factor out: ab(b - a)
Example 23: 2xyz - 4xy
Identify common factors: 2xy appears in both terms
Factor out: 2xy(z - 2)
Example 24: -12mn + 12m²n
Identify common factors: 12mn appears in both terms
Factor out: 12mn(-1 + m) or -12mn(1 - m)
Example 25: 6xy²z² - 3z²y
Identify common factors: 3z²y appears in both terms
Factor out: 3z²y(2xy - 1)
Factoring by Grouping: A Powerful Technique
For expressions with four terms, factoring by grouping is a powerful approach.
Step-by-Step Approach:
Group the terms into pairs
Factor out common terms from each pair
Identify the common binomial factor
Factor out the common binomial
Examples:
Example 26: 4x + 8 - ax - 2a
Group terms: (4x + 8) + (-ax - 2a)
Factor out common terms: 4(x + 2) - a(x + 2)
Identify common factor: (x + 2)
Factor out: (x + 2)(4 - a)
Example 27: 11x + 55 - ax - 5a
Group terms: (11x + 55) + (-ax - 5a)
Factor out common terms: 11(x + 5) - a(x + 5)
Identify common factor: (x + 5)
Factor out: (x + 5)(11 - a)
Example 28: ax + 5a - 4x - 20
Group terms: (ax + 5a) + (-4x - 20)
Factor out common terms: a(x + 5) - 4(x + 5)
Identify common factor: (x + 5)
Factor out: (x + 5)(a - 4)
Factoring Expressions with Brackets
Sometimes we need to expand expressions with brackets and then factor them.
Examples:
Example 29: 5x(x - 1) - a(x - 1)
Expand: 5x² - 5x - ax + a
Rearrange: 5x² - ax - 5x + a
Group terms: (5x² - ax) + (-5x + a)
Factor out common terms: x(5x - a) - 1(5x - a)
Identify common factor: (5x - a)
Factor out: (5x - a)(x - 1)
Example 30: bx + 2b + 3x + 6
Rearrange: bx + 3x + 2b + 6
Group terms: (bx + 3x) + (2b + 6)
Factor out common terms: x(b + 3) + 2(b + 3)
Identify common factor: (b + 3)
Factor out: (b + 3)(x + 2)
Special Factorization Cases
When One Term Has No Common Factor:
Sometimes, after factoring out common terms from groups, one of the resulting terms might be just "1".
Example 31: a(x + 2) + (x + 2)
Identify common factor: (x + 2)
Factor out: (x + 2)(a + 1)
Example 32: (x - 6)(1 - x)
This expression is already factored
We can simplify by noting that (1 - x) = -(x - 1)
Rewrite: -(x - 6)(x - 1)
Key Strategies for Successful Factorization
Always look for common factors first: Before trying other techniques, check if there's a common factor that can be factored out.
Be careful with signs: When factoring out negative terms, pay special attention to how signs change inside the brackets.
Group strategically: When factoring by grouping, try different groupings if the first attempt doesn't yield a common binomial factor.
Verify your factorization: Multiply your factored expression to check that it equals the original expression.
Practice regularly: Factorization skills improve with consistent practice across various types of expressions.
Common Mistakes to Avoid
Forgetting to change signs: When factoring out a negative term, remember that all signs inside the bracket must change.
Missing common factors: Always look carefully for all possible common factors before proceeding.
Incorrect grouping: When factoring by grouping, ensure that your grouping leads to a common binomial factor.
Forgetting to factor completely: Sometimes factored expressions can be factored further—always check if additional factorization is possible.
Building Your Factorization Skills
At HappyMath, we believe that factorization is a skill that develops with practice. Start with simpler expressions involving common factors, then progress to more complex expressions requiring grouping or special techniques.
With time and practice, you'll develop the mathematical intuition to recognize factorization patterns quickly, enabling you to tackle even the most challenging algebraic expressions with confidence.
Remember that factorization is both a technique and an art—there's often more than one correct approach, and the most elegant solution may not always be the most obvious one. Keep practicing, and you'll continue to refine your skills in this essential area of algebra.
