➕ Algebra 1 · Factoring
Difference of Squares: 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.
Settling up
Last week you paid 8 dollars for your friend’s lunch. This week your friend paid 8 dollars for yours. Now you are even: the amount you are owed and the amount you owe are the same size, in opposite directions, so they cancel. If your friend had paid only 5 dollars, you would not be even.
How it connects: When you multiply (a + b)(a − b), the middle terms are −ab and +ab. Same size, opposite signs, so they cancel and only a2 − b2 is left. The limit: cancelling needs equal sizes and opposite signs. In (a + b)(a + b) the middle terms have the same sign and add to 2ab. In (x + 3)(x − 2) they are −2x and +3x, which are different sizes, so +x is left over.
Dreams inside dreams
In the movie Inception, a team enters a person’s dream, and some missions go into a dream inside that dream, and then deeper still. To finish the job they go down one layer at a time.
How it connects: Factoring x4 − 16 works like that. The first layer is (x2 + 4)(x2 − 4). The factor x2 − 4 is itself a difference of squares, so you go one layer deeper: (x + 2)(x − 2). The factor x2 + 4 is a sum of squares, so it is the bottom layer and you stop. The limit: there is not always a deeper layer. x2 + 9 has nothing below it, and x2 − 7 has nothing below it either. Unlike a dream, you can always check by multiplying back up.
Don’t judge a book by its cover
This proverb means you should not decide what something is really like from its outward appearance alone. The expression 27x2 − 12 looks like it cannot be a difference of squares, because neither 27 nor 12 is a perfect square.
How it connects: Look inside by taking out the GCF: 27x2 − 12 = 3(9x2 − 4) = 3(3x + 2)(3x − 2). The limit: the proverb says to look deeper, not that every cover hides a surprise. 2x2 − 5 has no GCF to take out and is still not a difference of squares, and x2 + 9 really is as plain as it looks.
Hidden in plain sight
Something “hidden in plain sight” is right in front of you, but easy to miss because you are not looking for it. The expression m10 − 9 does not look like a difference of squares at first glance.
How it connects: But m10 = (m5)2 and 9 = 32, so m10 − 9 = (m5)2 − 32 = (m5 + 3)(m5 − 3). The squares are there if you halve the exponent. The limit: only an even exponent hides a square. m9 − 9 is not a difference of squares, because no whole-number power of m, squared, gives m9.
The brownie pan
A square pan of brownies is 7 pieces by 7 pieces, so 49 pieces. Someone eats a 3-by-3 block of 9 pieces from one corner, and 40 pieces are left in an L-shape. Cut the L into a strip 7 pieces wide and 4 tall (28 pieces) and a block 4 wide and 3 tall (12 pieces). Turn the block on its side and slide it to the end of the strip: now you have a rectangle 10 pieces wide and 4 tall.
How it connects: Here a = 7 and b = 3: 72 − 32 = 49 − 9 = 40, and (7 + 3)(7 − 3) = 10 × 4 = 40. The limit: the picture needs the eaten corner to be smaller than the pan and the pieces to be whole. The algebra is true for any numbers, even ones that do not fit in a pan.
The undo button
Many puzzle games have an undo button. Press it and you step back to the position before your last move, and redo takes you forward again. Multiplying (x + 5)(x − 5) gives x2 − 25. Factoring is the undo button: it takes x2 − 25 back to (x + 5)(x − 5).
How it connects: Checking a factorization is pressing redo: multiply the factors and see whether you land back on the start. The limit: an undo button always has an earlier move to go back to, but not every expression has factors. x2 + 25 was never made by multiplying two binomials with integer coefficients, so there is nothing to undo, and it is prime.
The club with a rule
Why was the sum of squares turned away from the Difference of Squares Club? It couldn’t make a difference.
How it connects: The club’s rule is the minus sign: x2 − 9 = (x + 3)(x − 3), while x2 + 9 is prime. The limit: the joke is about the pattern, not about sums being worse. A sum of squares is a perfectly good factor, as in x4 − 16 = (x2 + 4)(x + 2)(x − 2).
Memory help
FACTS
A checklist for factoring a difference of squares. F: take out any GCF First. A: Are there two terms, with one perfect square subtracted from another? C: Choose a and b, the things that were squared. T: write Two factors, (a + b)(a − b). S: Spot-check by expanding, and Scan the factors for anything that factors again. Say it as “FACTS.”
Root the first, root the last
Root the first, root the last; one plus, one minus, but you’re not done that fast. Expand to check: the middle is gone. If a middle term shows up, something went wrong. (This is for a minus sign between two perfect squares, a2 − b2. Take out any GCF first, and look again at each factor to see whether it factors more.)
Picture the L
Picture a big square with a small square bitten out of one corner. That L-shaped piece is a2 − b2. Cut it in two and slide the pieces together into a rectangle that is a + b wide and a − b tall. If you can picture the rectangle, you can write the two factors.
Root the number, halve the power
To find what was squared in a perfect-square term, take the square root of the number and cut each exponent in half. For 49x10: the root of 49 is 7 and half of 10 is 5, so it is (7x5)2. This only works when the number is a perfect square and the exponent is even.
A minus is not a square
x2 − 36 is not (x − 6)2. Expanding (x − 6)2 gives x2 − 12x + 36, which has a middle term. The factors of a difference of squares have opposite signs: (x + 6)(x − 6).
A plus sign stops the pattern
x2 + 64 is not (x + 8)(x − 8), because that product is x2 − 64. A two-term sum of squares like x2 + 64 is prime. Do not force it, and do not forget the coefficient: 25x2 − 64 is (5x + 8)(5x − 8), not (25x + 8)(25x − 8).