Factoring Techniques

Why This Matters

Factoring is the reverse of multiplication—and it's one of the most heavily tested skills in algebra and trigonometry. You're being tested on your ability to recognize patterns, apply formulas, and break down complex expressions into simpler pieces. Every technique you learn here connects directly to solving equations, simplifying rational expressions, and working with functions later in the course.

Don't just memorize the formulas—understand when each technique applies and why it works. The difference between a quick solution and a frustrating dead-end often comes down to pattern recognition. Learn to see the structure of an expression first, then choose your tool.

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Extracting Common Factors

Before attempting any other technique, always check for factors shared by every term. This simplifies the expression and often reveals hidden patterns.

Greatest Common Factor (GCF)

• Identify the largest factor (number and/or variable) that divides evenly into every term of the polynomial
• Factor out the GCF first—this reduces coefficients and exponents, making subsequent steps cleaner
• Check your work by distributing the GCF back through; you should get the original expression

Factoring by Grouping

• Group terms strategically (usually in pairs) so each group shares a common factor
• Factor each group separately, then extract the common binomial factor that emerges
• Best for four-term polynomials—if you see four terms, grouping should be your first instinct

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Recognizing Special Patterns

These formulas let you factor instantly once you recognize the structure. Memorize the forms—they appear constantly on exams.

Difference of Squares

• Pattern: $a^2 - b^2 = (a + b)(a - b)$—works only for subtraction of two perfect squares
• Both terms must be perfect squares—look for coefficients like 1, 4, 9, 16, 25 and even exponents on variables
• Does not apply to sums—$a^2 + b^2$ cannot be factored over the real numbers

Sum and Difference of Cubes

• Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$—note the minus in the trinomial
• Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$—note the plus in the trinomial
• Memory trick: "SOAP"—Same, Opposite, Always Positive describes the signs in order

Perfect Square Trinomials

• Pattern: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$
• Check the middle term—it must equal exactly $2ab$ (twice the product of the square roots of the outer terms)
• Faster than general trinomial methods when you spot the pattern; saves significant time on timed exams

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Factoring Trinomials

When special patterns don't apply, use systematic methods to factor expressions of the form $ax^2 + bx + c$.

Trinomials with Leading Coefficient 1

• Find two numbers that multiply to $c$ and add to $b$—these become your binomial constants
• Write directly: $x^2 + bx + c = (x + m)(x + n)$ where $mn = c$ and $m + n = b$
• Watch your signs—if $c$ is positive, both numbers have the same sign; if $c$ is negative, they have opposite signs

Trinomials with Leading Coefficient ≠ 1

• AC Method: multiply $a \cdot c$, find two numbers that multiply to $ac$ and add to $b$
• Split the middle term using those two numbers, then factor by grouping the resulting four-term expression
• Alternative: use trial and error with factor pairs of $a$ and $c$, but the AC method is more systematic

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Advanced Strategies

When basic techniques don't immediately apply, these methods help you see hidden structure.

Factoring by Substitution

• Replace a complex expression with a single variable (e.g., let $u = x^2$) to reveal a familiar pattern
• Factor the simpler expression in terms of $u$, then substitute back the original expression
• Especially useful for expressions like $x^4 - 5x^2 + 4$, which becomes $u^2 - 5u + 4$ with $u = x^2$

Complete Factorization

• Apply techniques in order: GCF first, then special patterns, then trinomial methods
• Factor completely means every factor is either prime (unfactorable) or a monomial
• Check each factor—after your first factorization, examine whether any factor can be broken down further

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Quick Reference Table

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Self-Check Questions

1. What is the first step you should always take before applying any other factoring technique?

2. How can you distinguish between a perfect square trinomial and a general trinomial that requires the AC method?

3. Compare and contrast the formulas for sum of cubes and difference of cubes—what stays the same, and what changes?

4. Given the expression $x^4 - 16$, which techniques would you need to apply to factor it completely? (Hint: it requires more than one step.)

5. If you encounter $6x^2 + 11x - 10$, explain why you would use the AC method rather than looking for a special pattern.