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Any polynomial factored, with the name of the pattern behind each step.
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The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
There are only about six things that ever happen when you factor a polynomial: a common factor comes out, a difference of two squares splits, a perfect square collapses into a bracket squared, a trinomial breaks into two brackets, four terms group in pairs, or a sum or difference of cubes opens up. Everything you will be asked to factor in school algebra is one of those, or one of those applied twice.
So the useful thing a calculator can tell you is not the answer - it is which pattern it used, and why that one. Every line here is named. When the same problem comes up in an exam you are not trying to remember what this page said; you are looking for the pattern it pointed at.
It works exactly, on whole-number and fractional coefficients alike, and it verifies itself: the last line multiplies the factors back out and shows you the polynomial you started with. That check is the answer to "how do I know this is right", and you can do it by hand on any answer, from any source, forever.
Write it the way you would on paper: x^2 - 5x + 6, 4x^2 - 12x + 9, x^3 - 8. Use ^ for powers. One letter, or two (x^2 - y^2 works).
The factored form is at the top. The numbered steps below name the pattern each one used - that is the part that transfers to the next question.
Every bracket of the form (ax - b) gives you a zero at b/a. That is why factoring solves equations: a product is zero exactly when one of its factors is.
The last line multiplies the factors back out. Cover it up, do it in your head, and then compare - that habit is worth more than the answer.
Six patterns cover school factoring. Count the terms first: it eliminates most of the list before you try anything.
| Pattern | Looks like | Factors to |
|---|---|---|
| Common factor | 6x² - 15x | 3x(2x - 5) |
| Difference of squares | 9x² - 4 | (3x - 2)(3x + 2) |
| Perfect square trinomial | 4x² - 12x + 9 | (2x - 3)² |
| Trinomial, leading 1 | x² - 5x + 6 | (x - 2)(x - 3) |
| Trinomial, leading not 1 | 2x² + 7x + 3 | (2x + 1)(x + 3) |
| Grouping (four terms) | x³ + 2x² + 3x + 6 | (x + 2)(x² + 3) |
| Difference of cubes | x³ - 8 | (x - 2)(x² + 2x + 4) |
| Sum of cubes | x³ + 27 | (x + 3)(x² - 3x + 9) |
| Sum of squares | x² + 9 | does not factor |
x^2 - 5x + 6Two numbers that multiply to 6 and add to -5: -2 and -3. So it is (x - 2)(x - 3). With a leading coefficient of 1 this is the whole method, and it is worth doing a dozen of them until the pair jumps out without a search.
4x^2 - 12x + 94x² is (2x)², 9 is 3², and the middle term is exactly -2 × 2x × 3. That makes it (2x - 3)². You could also grind it out as a trinomial and get (2x-3)(2x-3); recognising the square saves the work, which is the only reason patterns are worth learning.
9x^2 - 4The trap here is not seeing it, because the squares are not obvious: 9x² is (3x)² and 4 is 2². Then the pattern applies as usual and it splits into (3x - 2)(3x + 2). Two terms with a minus between them is always worth checking for this.
3x^2 - 27As written it is not a difference of squares - 3x² is not a square. Take the 3 out and 3(x² - 9) is, giving 3(x - 3)(x + 3). Skipping the common factor is the single most common way to conclude wrongly that something does not factor.
x^3 - 8x³ - 8 is x³ - 2³, so it is (x - 2)(x² + 2x + 4). The quadratic bracket never factors further over the rationals, which the page says rather than leaving you to wonder. Note the signs: the linear bracket takes the sign of the original, the quadratic one alternates.
