The keyword how to factor refers to the algebra process of rewriting an expression as a product of simpler expressions. Factoring is the reverse of expanding: it reveals structure, simplifies rational expressions, and is essential for solving polynomial equations.
Definition To factor an expression means to rewrite it as a product, for example: \[ x^2-9=(x-3)(x+3). \] A correct factorization multiplies back to the original expression.
Strategy: choose a factoring method
Factoring is most reliable when approached as a decision process: check the simplest patterns first, then move to more specialized methods.
| Expression form | What to try first | Typical result |
|---|---|---|
| Any polynomial | Greatest common factor (GCF) | Pull out a common number/variable factor |
| 4 terms (e.g., \(ax+ay+bx+by\)) | Factor by grouping | Common binomial factor |
| \(a^2-b^2\) | Difference of squares | \((a-b)(a+b)\) |
| \(a^2\pm 2ab+b^2\) | Perfect-square trinomial | \((a\pm b)^2\) |
| \(x^2+bx+c\) | Trinomial factoring | \((x+m)(x+n)\) |
| \(ax^2+bx+c\) with \(a\ne 1\) | \(ac\)-method (split and group) | \((px+q)(rx+s)\) |
Method 1: factor out the greatest common factor
The first step in how to factor is almost always to extract the GCF, because it simplifies what remains.
Example: \(6x^3-9x^2\)
The GCF is \(3x^2\), so:
\[ 6x^3-9x^2 = 3x^2(2x-3). \]
Method 2: factor by grouping
Example: \(x^3+3x^2+2x+6\)
- Group terms: \((x^3+3x^2)+(2x+6)\).
- Factor each group: \(x^2(x+3)+2(x+3)\).
- Factor the common binomial: \((x+3)(x^2+2)\).
\[ x^3+3x^2+2x+6=(x+3)(x^2+2). \]
Method 3: special products
Difference of squares
\[ a^2-b^2=(a-b)(a+b). \]
Example: \(9x^2-16=(3x)^2-4^2=(3x-4)(3x+4)\).
Perfect-square trinomials
\[ a^2+2ab+b^2=(a+b)^2, \qquad a^2-2ab+b^2=(a-b)^2. \]
Example: \(x^2+10x+25=x^2+2\cdot x\cdot 5+5^2=(x+5)^2\).
Method 4: factoring a quadratic trinomial
Quadratics are central in algebra. Two core cases are presented: \(a=1\) and \(a\ne 1\).
Case A: \(x^2+bx+c\)
Find integers \(m,n\) such that \(m+n=b\) and \(mn=c\). Then:
\[ x^2+bx+c=(x+m)(x+n). \]
Example: \(x^2+7x+12\)
Numbers \(3\) and \(4\) add to \(7\) and multiply to \(12\), so:
\[ x^2+7x+12=(x+3)(x+4). \]
Case B: \(ax^2+bx+c\) with \(a\ne 1\) (\(ac\)-method)
- Compute \(ac\).
- Find two numbers that multiply to \(ac\) and add to \(b\).
- Split the middle term and factor by grouping.
Example: \(6x^2+11x+3\)
Here \(a=6\), \(b=11\), \(c=3\), so \(ac=18\). Two numbers with product \(18\) and sum \(11\) are \(9\) and \(2\).
\[ 6x^2+11x+3=6x^2+9x+2x+3. \]
\[ =3x(2x+3)+1(2x+3)=(3x+1)(2x+3). \]
Visualization: factoring as area (trinomial to rectangle)
The area model below illustrates the factorization \(x^2+7x+12=(x+3)(x+4)\). A rectangle with side lengths \(x+3\) and \(x+4\) decomposes into four regions whose areas sum to the original trinomial.
Final checklist for how to factor
1 Factor out the GCF first.
2 Look for special products: difference of squares, perfect-square trinomials.
3 If four terms, try grouping.
4 If quadratic, use trinomial factoring (\(a=1\)) or the \(ac\)-method (\(a\ne 1\)).
5 Check by expanding to confirm the factorization matches the original expression.
Summary
The core idea of how to factor is to rewrite a polynomial as a product by applying a structured sequence of methods: extract the GCF, use grouping when appropriate, recognize special products, and factor quadratics using pattern recognition or the \(ac\)-method, then verify by expansion.