Special Products
Derive and apply the patterns for (a + b)², (a − b)², and (a + b)(a − b) = a² − b². Recognize these patterns to multiply special pairs quickly and verify that they follow from the distributive property.
A lesson of Algebra 1, in Polynomials, about 30 minutes.
By the end of this lesson, you will be able to recognize and expand the three patterns that come up constantly, spot them in reverse, and avoid the single most common algebra error there is.
Three Patterns Worth Recognizing
Every one of these is ordinary multiplication. The point is not a shortcut, it is recognition: spotting these shapes is what makes factoring possible later.
$$(a + b)^2 = a^2 + 2ab + b^2$$
$$(a - b)^2 = a^2 - 2ab + b^2$$
$$(a + b)(a - b) = a^2 - b^2$$
The first two are perfect squares. The third is the difference of two squares, and it is the strangest: the middle terms cancel and vanish completely.
Expand the third by hand once and it is obvious why:
$$(a+b)(a-b) = a^2 - ab + ab - b^2 = a^2 - b^2$$
$-ab$ and $+ab$ annihilate. That is the only pattern of the three where a term disappears, which is exactly what makes it recognizable in reverse.
The Middle Term Is the Whole Point
$$(x + 5)^2 \neq x^2 + 25$$
This is the most common error in all of algebra, and it is worth being precise about why.
$(x+5)^2$ means $(x+5)(x+5)$, which is four products, and the two middle ones do not cancel here, they ADD:
$$x^2 + 5x + 5x + 25 = x^2 + 10x + 25$$
The $10x$ is not decoration. Put $x = 1$ into both versions: the correct one gives $36$, and $x^2 + 25$ gives $26$. They are different by ten, which is the missing middle term.
Squaring a sum is not squaring the terms. Say $(x+5)^2 = (x+5)(x+5)$ out loud and expand it as a product until the pattern is automatic.
Example 1: A Perfect Square
Here is the whole method on one, worked all the way through. $$(x + 6)^2$$
- Write what the square actually means, as a product (x + 6)(x + 6)
- What is the first term of the answer? x^2
- The two middle products are both 6x. What do they come to? 12x
- What is the last term? 36
- Write the whole answer x^2 + 12x + 36
Worked Example 1: A Square with a Minus
$$(x - 4)^2$$
Worked Example 2: Difference of Two Squares
$$(x + 7)(x - 7)$$
Sort It: Which Pattern Is It
Reading Them Backwards
This is why the lesson exists. Recognizing these shapes in an ANSWER is how factoring starts.
$$x^2 - 25 \longrightarrow (x + 5)(x - 5)$$
Two terms, both squares, subtracted. That is a difference of squares every time, and it factors in one step with no trial and error.
$$x^2 + 12x + 36 \longrightarrow (x + 6)^2$$
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