Unit: Factorization Of Expressions
Chapter: Introduction to Factorization of Expressions
Reference: – What is Factorization, Factors of an Expression, Why Factor Expressions, Prime Factorization Review, Factorizing by Finding Common Factors, Greatest Common Factor (GCF), Factorizing by Grouping, Factorizing Quadratic Expressions (x² + bx + c), Factorizing Difference of Squares, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- What is Factorization
- How to Find the Greatest Common Factor (GCF) of an Expression
- How to Factor by Taking Out the Common Factor
- How to Factor by Grouping
- How to Factor Simple Quadratic Expressions
Introduction to Factorization of Expressions
Definition
Factorization (or factoring) is the process of writing an algebraic expression as a product of its factors. It is the reverse of expanding (multiplying out). When we factor an expression, we break it down into simpler expressions that multiply together to give the original expression.
When we factor an expression, we essentially ask:
"What expressions can I multiply together to get this expression?"
Just as numbers can be factored (12 = 3 × 4), algebraic expressions can also be factored.
Importance of Factorization
- Simplifies algebraic fractions
- Helps solve quadratic equations (zero product property)
- Reveals roots and intercepts of functions
- Makes complex expressions easier to work with
- Essential for higher algebra and calculus
Example
The expression 6x + 9 can be factored as 3(2x + 3). The factors are 3 and (2x + 3). If you multiply 3(2x + 3), you get back 6x + 9.
Subtopics
1. Factors of an Expression
A factor is an expression that divides another expression exactly (with no remainder).
Number Factors: 12 = 3 × 4, so 3 and 4 are factors of 12
Variable Factors: x² = x × x, so x and x are factors of x²
Expression Factors: 2x + 4 = 2(x + 2), so 2 and (x + 2) are factors of 2x + 4
2. Prime Factorization Review (Numbers)
Before factoring expressions, recall prime factorization of numbers. A prime number has exactly two factors: 1 and itself.
Example: 18 = 2 × 3 × 3 = 2 × 3²
3. Greatest Common Factor (GCF) of an Expression
The GCF of an algebraic expression is the largest factor that divides every term in the expression.
Steps to Find GCF:
Step 1: Find the GCF of the coefficients (numbers)
Step 2: For each variable, take the smallest exponent that appears in all terms
Step 3: Multiply these together
Example – Find GCF of 12x³ and 18x²
Coefficients: GCF of 12 and 18 is 6
Variables: smallest exponent of x is 2 (x²)
GCF = 6x²
4. Factorizing by Taking Out the Common Factor (Factoring Out GCF)
This is the simplest method. Identify the GCF of all terms and write it outside parentheses.
Steps:
Step 1: Find the GCF of all terms
Step 2: Divide each term by the GCF
Step 3: Write the expression as GCF × (quotients)
Example 1: Factor 6x + 9
GCF of 6 and 9 is 3
6x ÷ 3 = 2x, 9 ÷ 3 = 3
6x + 9 = 3(2x + 3)
Example 2: Factor 8x² – 12x
GCF of 8 and 12 is 4; smallest exponent of x is 1 → GCF = 4x
8x² ÷ 4x = 2x, -12x ÷ 4x = -3
8x² – 12x = 4x(2x – 3)
Example 3: Factor 5x²y – 10xy²
GCF of 5 and 10 is 5; smallest exponent of x is 1; smallest exponent of y is 1 → GCF = 5xy
5x²y ÷ 5xy = x, -10xy² ÷ 5xy = -2y
5x²y – 10xy² = 5xy(x – 2y)
5. Factorizing by Grouping
Used when an expression has four terms. Group terms that have common factors, factor each group, then factor out the common binomial.
Steps:
Step 1: Group the terms into two pairs
Step 2: Factor out the GCF from each pair
Step 3: Look for a common binomial factor
Step 4: Factor out the binomial
Example: Factor x³ + 2x² + 3x + 6
Group: (x³ + 2x²) + (3x + 6)
Factor each group: x²(x + 2) + 3(x + 2)
Common binomial (x + 2): (x + 2)(x² + 3)
Answer: (x + 2)(x² + 3)
6. Factorizing Quadratic Expressions (x² + bx + c)
A quadratic expression in the form x² + bx + c (coefficient of x² is 1) factors into (x + p)(x + q) where p + q = b and p × q = c.
Steps:
Step 1: Find two numbers p and q such that p + q = b and p × q = c
Step 2: Write the factors as (x + p)(x + q)
Example 1: Factor x² + 5x + 6
Find p + q = 5, p × q = 6 → p = 2, q = 3 (or 3 and 2)
x² + 5x + 6 = (x + 2)(x + 3)
Example 2: Factor x² – 7x + 12
p + q = -7, p × q = 12 → p = -3, q = -4 (since -3 + -4 = -7, -3 × -4 = 12)
x² – 7x + 12 = (x – 3)(x – 4)
Example 3: Factor x² + 2x – 15
p + q = 2, p × q = -15 → p = 5, q = -3 (since 5 + -3 = 2, 5 × -3 = -15)
x² + 2x – 15 = (x + 5)(x – 3)
7. Factorizing Difference of Squares
A special case: a² – b² = (a – b)(a + b)
Example 1: Factor x² – 16 = x² – 4² = (x – 4)(x + 4)
Example 2: Factor 9x² – 25 = (3x)² – 5² = (3x – 5)(3x + 5)
Example 3: Factor 4y² – 49 = (2y)² – 7² = (2y – 7)(2y + 7)
Note: Sum of squares (a² + b²) does NOT factor over real numbers.
Solved Examples
Example 1 – Factoring Out GCF: Factor 15x³ – 25x²
Solution: GCF of 15 and 25 is 5; smallest exponent of x is 2 → GCF = 5x²
15x³ ÷ 5x² = 3x, -25x² ÷ 5x² = -5
Answer: 5x²(3x – 5)
Example 2 – Factoring Quadratic: Factor x² + 8x + 15
Solution: p + q = 8, p × q = 15 → p = 3, q = 5
Answer: (x + 3)(x + 5)
Example 3 – Difference of Squares: Factor 16x² – 81
Solution: 16x² = (4x)², 81 = 9²
Answer: (4x – 9)(4x + 9)
Example 4 – Factor by Grouping: Factor x³ – 3x² + 2x – 6
Solution: Group: (x³ – 3x²) + (2x – 6)
Factor each: x²(x – 3) + 2(x – 3)
Common binomial: (x – 3)(x² + 2)
Answer: (x – 3)(x² + 2)
Common Mistakes to Avoid
Mistake 1 – Factoring incompletely
4x + 8 = 2(2x + 4) is not fully factored because 2x + 4 still has a common factor of 2.
Correct understanding: 4x + 8 = 4(x + 2) – factor out the GCF completely.
Mistake 2 – Sign errors when factoring quadratics
For x² – 5x + 6 = (x + 2)(x – 3)? Check: x² – 3x + 2x – 6 = x² – x – 6, not correct.
Correct understanding: (-2) + (-3) = -5, (-2) × (-3) = 6 → (x – 2)(x – 3).
Mistake 3 – Forgetting that not all quadratics factor
x² + x + 1 does not factor over integers.
Correct understanding: Check if you can find integer p and q with p + q = b and p × q = c.
Mistake 4 – Applying difference of squares to sum of squares
x² + 25 is NOT factorable as (x + 5)(x – 5) because that gives x² – 25.
Correct understanding: a² + b² does NOT factor over real numbers.
Mistake 5 – Leaving a negative GCF
-4x – 8 = -4(x + 2) is correct, but -4(x + 2) is often preferred.
Correct understanding: Always factor out the negative sign if the first term is negative.
Mistake 6 – Grouping terms incorrectly
For grouping, terms must be grouped so that each pair has a common factor.
Correct understanding: Try different groupings if the first grouping does not work.
Quick Reference Summary
Factorization: Writing an expression as a product of its factors
GCF Method: Find GCF of all terms, factor it out: GCF × (remaining expression)
Grouping Method: For 4 terms – group, factor each group, factor common binomial
Quadratic (x² + bx + c): Find p, q such that p + q = b, p × q = c → (x + p)(x + q)
Difference of Squares: a² – b² = (a – b)(a + b)
Sum of Squares: a² + b² does NOT factor over real numbers
Check Your Work: Multiply the factors to see if you get back the original expression.