➕ Algebra 1 · Exponents and Polynomials
Special Products of Binomials: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
The cookie tray
A bakery sells cookies on a square tray with 11 rows of 11 cookies. Counting row by row is slow. Instead, see the tray as a 10-by-10 block of 100 cookies, plus a strip of 10 along one side, plus a strip of 10 along the other side, plus 1 cookie in the corner.
How it connects: (10 + 1)2 = 102 + 2(10)(1) + 12 = 100 + 20 + 1 = 121. The two strips of 10 are the middle term 2ab, and the corner cookie is b2. Someone who writes 112 = 102 + 12 gets 101 and loses the 20 cookies in the strips. The limit: the picture needs positive counts, but the pattern works for every number.
Trimming a gym mat
A square gym mat is 12 feet on each side. The coach trims 2 feet off two neighboring sides, so the mat is now 10 feet by 10 feet. Each trimmed strip is 12 feet by 2 feet, which is 24 square feet. But the 2-by-2 corner belongs to both strips, so the total removed is 24 + 24 − 4 = 44 square feet.
How it connects: (12 − 2)2 = 122 − 2(12)(2) + 22 = 144 − 48 + 4 = 100. Subtracting both strips removes the corner twice, so the +b2 gives it back once. That is why the last term of (a − b)2 is positive. The limit: the picture needs a > b > 0, but the pattern holds for every number.
Stretch and squash
In a game’s level editor, a square tile is 20 pixels wide and 20 pixels tall. You stretch it 3 pixels wider and squash it 3 pixels shorter, so it is 23 by 17. It feels like you added as much as you took away, but the area went from 20 × 20 = 400 to 23 × 17 = 391, which is 9 less.
How it connects: (20 + 3)(20 − 3) = 202 − 32 = 391. The Outer and Inner products, −60 and +60, cancel, and only −b2 is left. The limit: this works because the same amount, 3, is added and removed. Stretching by 3 and squashing by 2 gives (20 + 3)(20 − 2) = 414, which needs a middle term.
Haste makes waste
This saying means that rushing a job usually leads to mistakes you have to redo, so you waste time instead of saving it.
How it connects: Writing (a + b)2 = a2 + b2 in a hurry feels fast, but the answer is wrong. Take one extra line to write the repeated factor, (a + b)(a + b), and multiply every term by every term. The limit: the lesson is to slow down at that one step, not to multiply out everything forever. Once you know the pattern, (a + b)2 = a2 + 2ab + b2 is both fast and correct.
Cancel each other out
When two things “cancel each other out,” they have opposite effects of the same size, so nothing is left. Gaining 5 points and then losing 5 points cancel each other out.
How it connects: In (a + b)(a − b) the outer and inner products are −ab and +ab. They cancel each other out, so there is no middle term: a2 − b2. The limit: they cancel only when the two factors use the same two terms. In (a + b)(a − c) the middle terms are −ac and +ab, which do not cancel unless b = c (or a = 0).
The cross term
Why is the cross term always so cross? Everybody squaring a + b keeps leaving it out.
How it connects: The middle term 2ab is called the cross term because it comes from multiplying across the two different terms, a × b and b × a. Writing (a + b)2 = a2 + b2 leaves it out. The limit: in (a + b)(a − b) the Outer and Inner products really do cancel, so there is no cross term to put back in. Do not add one there.
Memory help
S-D-S
For a binomial multiplied by itself: Square the first term, Double the product of the two terms, Square the last term. Use each term with its own sign. For (3m − 2)2 the terms are 3m and −2: square the first to get 9m2, double the product 2(3m)(−2) = −12m, and square the last, (−2)2 = 4. The answer is 9m2 − 12m + 4. S-D-S fits only a binomial squared. It does not fit (a + b)(a − b).
Plus-minus twins
Same two terms with a plus and a minus: the middle cancels, there is no fuss. Square the term that keeps its sign, then subtract the square of the one that flips. Two terms are left, and it is over fast. (In (x + 3)(3 − x) the 3 keeps its sign, so the answer is 9 − x2.)
The cut square
Draw a square, cut it once each way, and label the four pieces a2, ab, ab and b2. The two squares sit on one diagonal and the two matching rectangles on the other. If your answer shows only the two squares, the rectangles are missing.
The missing rectangles
(n + 14)2 is not n2 + 196. The two rectangles add 28n, so (n + 14)2 = n2 + 28n + 196. At n = 3 the left side is 289, but n2 + 196 gives only 205.
A minus in the middle, a plus at the end
(c − 7)2 = c2 − 14c + 49. Here the middle term takes the minus sign from the binomial, but the last term is (−7)2 = +49. It is not c2 − 49 (that is the answer to (c + 7)(c − 7)), and it is not c2 − 14c − 49.
Cut and slide
Sketch a square with a small square missing from one corner. Cut the L-shape into two rectangles, turn one, and slide them together into one long rectangle. That is a2 − b2 becoming (a + b)(a − b).