How To Factor An Expression Completely

Factoring an expression completely is a fundamental skill in algebra, allowing you to simplify complex expressions and solve equations more efficiently. It involves breaking down an expression into its prime factors, much like finding the prime factors of a number. This article will guide you through the process with detailed explanations and examples, covering various techniques and strategies to help you master this essential algebraic skill.

Understanding the Basics of Factoring

Factoring is the reverse process of expanding expressions. When you expand, you multiply terms together to get a larger expression. When you factor, you break down a larger expression into smaller, multiplied terms. The goal is to express a given expression as a product of its factors.

Key Concepts

• Factor: A number or expression that divides another number or expression evenly (i.e., with no remainder).
• Prime Factor: A factor that cannot be factored further.
• Greatest Common Factor (GCF): The largest factor that divides two or more numbers or expressions.
• Expression: A combination of numbers, variables, and operations.
• Equation: A statement that two expressions are equal.

Why is Factoring Important?

Factoring is essential for several reasons:

• Simplifying Expressions: Factoring can make complex expressions easier to understand and work with.
• Solving Equations: Many algebraic equations, especially quadratic equations, can be solved by factoring.
• Finding Roots: Factoring helps in finding the roots (or zeros) of a polynomial function.
• Calculus: Factoring is a crucial skill in calculus for simplifying derivatives and integrals.

Techniques for Factoring Expressions

Several techniques can be used to factor expressions completely. Here are some of the most common and effective methods.

1. Factoring Out the Greatest Common Factor (GCF)

The first step in factoring any expression is to look for the greatest common factor (GCF) of all the terms. The GCF is the largest factor that divides evenly into each term in the expression.

Steps to Factor Out the GCF:

1. Identify the GCF: Determine the greatest common factor of the coefficients and variables in the expression.
2. Divide Each Term by the GCF: Divide each term in the expression by the GCF.
3. Write the Factored Expression: Write the GCF outside the parentheses, followed by the result of the division inside the parentheses.

Example 1: Factoring the GCF

Factor the expression: 12x^3 + 18x^2 - 24x

1. Identify the GCF:
   • Factors of 12: 1, 2, 3, 4, 6, 12
   • Factors of 18: 1, 2, 3, 6, 9, 18
   • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
   • The greatest common factor of 12, 18, and 24 is 6.
   • The greatest common factor of x^3, x^2, and x is x.
   • Therefore, the GCF of the entire expression is 6x.
2. Divide Each Term by the GCF:
   • (12x^3) / (6x) = 2x^2
   • (18x^2) / (6x) = 3x
   • (-24x) / (6x) = -4
3. Write the Factored Expression:
   • 6x(2x^2 + 3x - 4)

Example 2: Factoring the GCF with Multiple Variables

Factor the expression: 15a^4b^2 - 25a^3b^3 + 30a^2b^4

1. Identify the GCF:
   • Factors of 15: 1, 3, 5, 15
   • Factors of 25: 1, 5, 25
   • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
   • The greatest common factor of 15, 25, and 30 is 5.
   • The greatest common factor of a^4, a^3, and a^2 is a^2.
   • The greatest common factor of b^2, b^3, and b^4 is b^2.
   • Therefore, the GCF of the entire expression is 5a^2b^2.
2. Divide Each Term by the GCF:
   • (15a^4b^2) / (5a^2b^2) = 3a^2
   • (-25a^3b^3) / (5a^2b^2) = -5ab
   • (30a^2b^4) / (5a^2b^2) = 6b^2
3. Write the Factored Expression:
   • 5a^2b^2(3a^2 - 5ab + 6b^2)

2. Factoring by Grouping

Factoring by grouping is used when you have an expression with four or more terms that don't have an obvious GCF for all terms. This method involves grouping terms and factoring out common factors from each group.

Steps for Factoring by Grouping:

1. Group the Terms: Arrange the terms into two or more groups.
2. Factor Each Group: Factor out the GCF from each group.
3. Identify the Common Binomial Factor: Look for a common binomial factor in the factored groups.
4. Factor Out the Common Binomial Factor: Factor out the common binomial factor from the expression.

Example 1: Factoring by Grouping

Factor the expression: ax + ay + bx + by

1. Group the Terms:
   • (ax + ay) + (bx + by)
2. Factor Each Group:
   • a(x + y) + b(x + y)
3. Identify the Common Binomial Factor:
   • The common binomial factor is (x + y).
4. Factor Out the Common Binomial Factor:
   • (x + y)(a + b)

Example 2: Factoring by Grouping with Rearrangement

Factor the expression: 3x^2 - 12x - 2x + 8

1. Group the Terms:
   • (3x^2 - 12x) + (-2x + 8)
2. Factor Each Group:
   • 3x(x - 4) - 2(x - 4)
3. Identify the Common Binomial Factor:
   • The common binomial factor is (x - 4).
4. Factor Out the Common Binomial Factor:
   • (x - 4)(3x - 2)

3. Factoring Trinomials

A trinomial is a polynomial with three terms. Factoring trinomials is a crucial skill, especially when dealing with quadratic expressions.

Factoring Trinomials of the Form x^2 + bx + c

Steps for Factoring x^2 + bx + c:

1. Find Two Numbers: Find two numbers that multiply to c and add up to b.
2. Write the Factored Form: Write the trinomial as a product of two binomials using the numbers found in step 1.

Example 1: Factoring x^2 + bx + c

Factor the trinomial: x^2 + 7x + 12

1. Find Two Numbers:
   • We need two numbers that multiply to 12 and add up to 7.
   • The numbers are 3 and 4 because 3 * 4 = 12 and 3 + 4 = 7.
2. Write the Factored Form:
   • (x + 3)(x + 4)

Example 2: Factoring x^2 + bx + c with Negative Numbers

Factor the trinomial: x^2 - 5x - 14

1. Find Two Numbers:
   • We need two numbers that multiply to -14 and add up to -5.
   • The numbers are -7 and 2 because -7 * 2 = -14 and -7 + 2 = -5.
2. Write the Factored Form:
   • (x - 7)(x + 2)

Factoring Trinomials of the Form ax^2 + bx + c

Factoring trinomials where a is not equal to 1 requires a bit more work. There are a couple of common methods: the trial and error method and the AC method.

Method 1: Trial and Error

1. List Possible Factors: List possible factors of a and c.
2. Test Combinations: Test different combinations of factors until you find a combination that gives you the correct middle term bx.
3. Write the Factored Form: Write the trinomial as a product of two binomials.

Example 1: Factoring ax^2 + bx + c using Trial and Error

Factor the trinomial: 2x^2 + 5x + 3

1. List Possible Factors:
   • Factors of 2: 1, 2
   • Factors of 3: 1, 3
2. Test Combinations:
   • Try (2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3 (Incorrect)
   • Try (2x + 3)(x + 1) = 2x^2 + 2x + 3x + 3 = 2x^2 + 5x + 3 (Correct)
3. Write the Factored Form:
   • (2x + 3)(x + 1)

Method 2: The AC Method

1. Multiply a and c: Multiply the coefficient of x^2 (a) by the constant term (c).
2. Find Two Numbers: Find two numbers that multiply to ac and add up to b.
3. Rewrite the Middle Term: Rewrite the middle term bx using the two numbers found in step 2.
4. Factor by Grouping: Factor the resulting expression by grouping.

Example 2: Factoring ax^2 + bx + c using the AC Method

Factor the trinomial: 3x^2 - 10x + 8

1. Multiply a and c:
   • a = 3, c = 8, so ac = 3 * 8 = 24
2. Find Two Numbers:
   • We need two numbers that multiply to 24 and add up to -10.
   • The numbers are -6 and -4 because -6 * -4 = 24 and -6 + -4 = -10.
3. Rewrite the Middle Term:
   • 3x^2 - 6x - 4x + 8
4. Factor by Grouping:
   • (3x^2 - 6x) + (-4x + 8)
   • 3x(x - 2) - 4(x - 2)
   • (x - 2)(3x - 4)

4. Factoring Special Products

Certain types of expressions have specific patterns that make them easier to factor. These are known as special products.

Difference of Squares: a^2 - b^2

The difference of squares can be factored as:

a^2 - b^2 = (a + b)(a - b)

Example 1: Factoring Difference of Squares

Factor the expression: x^2 - 25

1. Identify a and b:
   • a^2 = x^2, so a = x
   • b^2 = 25, so b = 5
2. Apply the Formula:
   • x^2 - 25 = (x + 5)(x - 5)

Example 2: Factoring Difference of Squares with Coefficients

Factor the expression: 4x^2 - 9y^2

1. Identify a and b:
   • a^2 = 4x^2, so a = 2x
   • b^2 = 9y^2, so b = 3y
2. Apply the Formula:
   • 4x^2 - 9y^2 = (2x + 3y)(2x - 3y)

Perfect Square Trinomials: a^2 + 2ab + b^2 and a^2 - 2ab + b^2

Perfect square trinomials can be factored as:

• a^2 + 2ab + b^2 = (a + b)^2
• a^2 - 2ab + b^2 = (a - b)^2

Example 1: Factoring Perfect Square Trinomial

Factor the expression: x^2 + 6x + 9

1. Identify a and b:
   • a^2 = x^2, so a = x
   • b^2 = 9, so b = 3
2. Check the Middle Term:
   • 2ab = 2 * x * 3 = 6x (Matches the middle term)
3. Apply the Formula:
   • x^2 + 6x + 9 = (x + 3)^2

Example 2: Factoring Perfect Square Trinomial with Subtraction

Factor the expression: 4x^2 - 12x + 9

1. Identify a and b:
   • a^2 = 4x^2, so a = 2x
   • b^2 = 9, so b = 3
2. Check the Middle Term:
   • -2ab = -2 * 2x * 3 = -12x (Matches the middle term)
3. Apply the Formula:
   • 4x^2 - 12x + 9 = (2x - 3)^2

Sum and Difference of Cubes: a^3 + b^3 and a^3 - b^3

The sum and difference of cubes can be factored as:

• a^3 + b^3 = (a + b)(a^2 - ab + b^2)
• a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Example 1: Factoring Sum of Cubes

Factor the expression: x^3 + 8

1. Identify a and b:
   • a^3 = x^3, so a = x
   • b^3 = 8, so b = 2
2. Apply the Formula:
   • x^3 + 8 = (x + 2)(x^2 - 2x + 4)

Example 2: Factoring Difference of Cubes

Factor the expression: 27x^3 - 1

1. Identify a and b:
   • a^3 = 27x^3, so a = 3x
   • b^3 = 1, so b = 1
2. Apply the Formula:
   • 27x^3 - 1 = (3x - 1)(9x^2 + 3x + 1)

Steps to Completely Factor an Expression

To completely factor an expression, follow these steps:

1. Look for a GCF: Always start by factoring out the greatest common factor (GCF) from all terms.
2. Check for Special Products: Determine if the expression is a difference of squares, a perfect square trinomial, or a sum/difference of cubes.
3. Factor Trinomials: If the expression is a trinomial, use the appropriate method (trial and error or AC method).
4. Factor by Grouping: If the expression has four or more terms and no obvious GCF, try factoring by grouping.
5. Check for Further Factoring: After each step, check to see if any of the factors can be factored further.
6. Write the Completely Factored Expression: Ensure that the expression is written as a product of prime factors.

Example: Factoring Completely

Factor the expression completely: 2x^3 - 8x

1. Look for a GCF:
   • The GCF of 2x^3 and -8x is 2x.
   • 2x(x^2 - 4)
2. Check for Special Products:
   • x^2 - 4 is a difference of squares.
   • x^2 - 4 = (x + 2)(x - 2)
3. Write the Completely Factored Expression:
   • 2x(x + 2)(x - 2)

Advanced Factoring Techniques

Some expressions may require more advanced techniques to factor completely.

Factoring by Substitution

Sometimes, an expression can be simplified by substituting a part of it with a single variable.

Example:

Factor the expression: (x^2 + 2x)^2 - 3(x^2 + 2x) - 4

1. Substitute:
   • Let y = x^2 + 2x
   • The expression becomes: y^2 - 3y - 4
2. Factor the New Expression:
   • y^2 - 3y - 4 = (y - 4)(y + 1)
3. Substitute Back:
   • Replace y with x^2 + 2x
   • (x^2 + 2x - 4)(x^2 + 2x + 1)
4. Check for Further Factoring:
   • x^2 + 2x + 1 = (x + 1)^2
   • The first quadratic, x^2 + 2x - 4, cannot be factored further using integers.
5. Write the Completely Factored Expression:
   • (x^2 + 2x - 4)(x + 1)^2

Factoring Expressions with Rational Exponents

Expressions with rational exponents can also be factored using similar techniques.

Example:

Factor the expression: x^(2/3) - 5x^(1/3) + 6

1. Substitute:
   • Let y = x^(1/3)
   • The expression becomes: y^2 - 5y + 6
2. Factor the New Expression:
   • y^2 - 5y + 6 = (y - 2)(y - 3)
3. Substitute Back:
   • Replace y with x^(1/3)
   • (x^(1/3) - 2)(x^(1/3) - 3)

Common Mistakes to Avoid

• Not Factoring Completely: Always ensure that each factor cannot be factored further.
• Incorrectly Applying Formulas: Double-check the formulas for special products before applying them.
• Forgetting the GCF: Always look for the GCF first before applying other factoring techniques.
• Sign Errors: Pay close attention to signs when factoring, especially with negative numbers.
• Stopping Too Early: Make sure you have factored all the way down to prime factors.

Conclusion

Factoring expressions completely is a critical skill in algebra. By mastering techniques such as factoring out the GCF, factoring by grouping, factoring trinomials, and recognizing special products, you can simplify complex expressions and solve equations more efficiently. Remember to always check for further factoring and avoid common mistakes to ensure complete and accurate results. With practice, you'll become proficient in factoring any algebraic expression that comes your way.