Factoring Special Cases
Factor perfect square trinomials (a² + 2ab + b² and a² − 2ab + b²) and differences of squares (a² − b²). Recognize these forms quickly and apply factoring formulas directly.
A lesson of Algebra 1, in Factoring Polynomials, about 35 minutes.
By the end of this lesson, you will be able to recognize a difference of squares and a perfect square trinomial on sight, factor both in one step, and know which lookalikes do not factor at all.
Two Patterns to Recognize Instantly
You expanded these in the Polynomials unit. Now you read them backwards.
$$a^2 - b^2 = (a + b)(a - b)$$
$$a^2 + 2ab + b^2 = (a + b)^2$$
$$a^2 - 2ab + b^2 = (a - b)^2$$
These would all yield to the AC method eventually. The point is that you do not have to run it: recognizing the shape gives the answer in one step, with no search at all.
$$x^2 - 49 = (x + 7)(x - 7)$$
$$x^2 + 10x + 25 = (x + 5)^2$$
Recognition beats method here, and on a timed test that difference is substantial.
Difference of Squares: Two Terms, Both Squares, Subtracted
Three conditions, all of them checkable at a glance.
$$4x^2 - 25$$
Two terms with nothing in the middle. Both are perfect squares: $4x^2 = (2x)^2$ and $25 = 5^2$. Subtracted.
$$4x^2 - 25 = (2x + 5)(2x - 5)$$
Take the square root of each term and write the two brackets, one plus and one minus. The middle terms cancel when you expand, which is why there was no middle term to start with.
The squares you should know on sight: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$. And a variable is a square when its power is even, so $x^6 = (x^3)^2$.
A sum of squares does not factor. $x^2 + 25$ has no real factorization, and this is the one asymmetry that catches people. There is no way for two brackets to multiply out with a canceling middle AND a positive constant.
Sort It: Difference of Squares or Not
Example 1: A Difference of Squares
Here is every step on one, worked all the way through. $$16x^2 - 81$$
- How many terms are there? 2
- What is the square root of the first term? 4x
- What is the square root of 81? 9
- Write the factored form (4x + 9)(4x - 9)
Perfect Square Trinomials
Three terms, and two of them squares.
$$x^2 + 12x + 36$$
First and last terms are perfect squares, here $x^2$ and $6^2$. The middle term is twice the product of their roots, here $2 \times x \times 6 = 12x$. That is the test, and both parts have to pass.
$$x^2 + 12x + 36 = (x + 6)^2$$
The middle term's sign becomes the bracket's sign. A negative middle gives $(x - 6)^2$, and the last term stays positive either way because it is a square.
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