Factoring Polynomials: Step-by-Step Guide
Factoring means rewriting an expression as a product of simpler expressions, like writing 12 as 3 x 4. The best plan is: first pull out the greatest common factor (GCF), the biggest thing that all the terms share. Then look for special patterns like a difference of squares. For a polynomial with four terms, try grouping.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Pull out the GCF. Find the biggest number and the most letters that all the terms share. Write it in front and divide each term by it.\[ax^{n} + bx^{m} = x^{\min(n,m)}\left(ax^{n-\min(n,m)} + bx^{m-\min(n,m)}\right)\]
- Look for a special pattern. Check whether what is left is a difference of squares (like x^2 - 4), a perfect-square trinomial, or a sum or difference of cubes. If so, use the matching rule.\[a^{2}-b^{2}=(a-b)(a+b),\quad a^{3}\pm b^{3}=(a\pm b)(a^{2}\mp ab+b^{2})\]
- Try grouping for four terms. Split the four terms into two pairs. Pull a common factor out of each pair, and then pull out the matching group they share.\[ax+ay+bx+by = a(x+y)+b(x+y) = (a+b)(x+y)\]
- Factor each piece all the way. Look at each factor. If it can still be factored, keep going until every piece is as simple as it can get.\[P(x)=c\cdot f_{1}(x)\cdot f_{2}(x)\cdots f_{k}(x)\]
- Check by multiplying. Multiply the factors back together. You should get the original polynomial.\[f_{1}(x)\cdot f_{2}(x) \stackrel{?}{=} P(x)\]
Worked example
Factor completely: 2x^3 - 8x
Step 1: Find the GCF. Each term has a factor of 2 and at least one x, so the GCF is 2x. Pull it out: 2x^3 - 8x = 2x(x^2 - 4).
Step 2: Notice that x^2 - 4 is a difference of squares, because 4 = 2^2. So it is x^2 - 2^2.
Step 3: Use the rule a^2 - b^2 = (a - b)(a + b): x^2 - 4 = (x - 2)(x + 2).
Step 4: Put all the pieces together: 2x^3 - 8x = 2x(x - 2)(x + 2).
Step 5: None of these factors can be broken down more, so we are done.
Check: Multiply back. 2x(x - 2)(x + 2) = 2x(x^2 - 4) = 2x^3 - 8x. It matches the original.
Tips and common mistakes
- Always look for the GCF first. It is the step people miss most, and it makes everything else easier.
- Learn these patterns: a^2 - b^2 = (a - b)(a + b), and a^2 + 2ab + b^2 = (a + b)^2.
- A sum of squares like x^2 + 9 cannot be factored with real numbers. Do not try to force it.
- After you factor once, look at each piece again. A common mistake is to stop at x^2 - 4 when it can still become (x - 2)(x + 2).
- For four terms, try grouping. If the first way does not work, switch the order of the terms and try again.
- If you pull out a negative from a group, every sign inside flips.
- Always multiply your answer back out to check. It catches sign mistakes quickly.