Step 1: Factor out the GCF

Key idea

The GCF is the largest factor shared by every term. Factor it out using the distributive property: \[ ab+ac=a(b+c). \] After you factor out the GCF, check again: the remaining factor may still be factorable.

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Summary

• Always factor out the GCF first.
• Then check for patterns or more factoring in what remains.

Step 2: Use common factoring patterns

Key idea

Two patterns show up constantly: \[ a^2-b^2=(a-b)(a+b) \] \[ a^2\pm2ab+b^2=(a\pm b)^2 \] If you see a binomial that looks like “square minus square,” use difference of squares. If you see a trinomial with a first and last term that are perfect squares and a middle term that matches \(\pm2ab\), it’s a perfect square trinomial.

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Summary

• Difference of squares: \(a^2-b^2=(a-b)(a+b)\).
• Perfect square trinomials: \(a^2\pm2ab+b^2=(a\pm b)^2\).

Step 3: Factor trinomials

Key idea

For \(x^2+bx+c\), find two numbers that multiply to \(c\) and add to \(b\). For \(ax^2+bx+c\), one reliable method is the \(ac\) method: find numbers that multiply to \(ac\) and add to \(b\), split the middle term, then factor by grouping.

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Summary

• For \(x^2+bx+c\): two numbers multiply to \(c\) and add to \(b\).
• For \(ax^2+bx+c\): use the \(ac\) idea, then group, then factor completely.

Factoring by grouping (four terms)

Key idea

Grouping works when you can split a polynomial into two groups that share a common factor, often a common binomial. A reliable process: (1) Group into pairs, (2) factor the GCF from each pair, (3) factor out the common binomial.

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Summary

• Grouping is best for four terms.
• After factoring out a common binomial, keep factoring (like factoring \(x^2-1\)).

Factor completely: repeated patterns and identities

Key idea

“Factor completely” means you keep going until nothing factors further over the integers. A common situation is a difference of squares inside a factor: \[ x^4-1=(x^2)^2-1^2=(x^2-1)(x^2+1), \] and then \(x^2-1\) factors again. Also remember: \[ x^3-1=(x-1)(x^2+x+1). \]

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Summary

• When you factor, always ask: “Can any factor be factored again?”
• Know the big identities: difference of squares and \(x^3-1\).

Check your factoring and solve equations

Key idea

The fastest way to verify a factorization is to multiply your factors and confirm you get the original expression. To solve a factored equation, use the zero product property: \[ (x-7)(x+7)=0 \Rightarrow x=7 \text{ or } x=-7. \]

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Summary

• Multiply factors to check your work.
• Use the zero product property to solve factored equations.

Why factoring techniques matter

Where factoring shows up

• Simplifying rational expressions: factoring lets you cancel common factors safely (with domain restrictions).
• Solving quadratics: factoring turns a quadratic into two linear equations.
• Graphs and intercepts: factored form shows zeros and x-intercepts quickly.
• Modeling and geometry: area/volume expressions often factor to reveal structure.

Worked example: simplify using factoring

Final check

Final recap

• Use the same order every time: GCF → patterns → trinomials → grouping → factor completely.
• Common patterns: difference of squares, perfect square trinomials, and \(x^3-1\).
• Check by multiplying, then use factoring to solve equations and simplify expressions.