Numbers & Functions · 4 Polynomials, Factoring, and the Quadratic Formula

4.5 A factoring toolkit you should own

Some factorings come up so often you should know them on sight. We’ll build the toolkit by multiplying out, because watching the expansion is the most honest proof that the identity is true.

1. Difference of squares. The most useful identity in elementary algebra: a2 −b2 = (a −b)(a + b).

Multiply the right side out: (a −b)(a + b) = a2 + ab −ba −b2. The two middle terms cancel, leaving a2 −b2. So x2 −9 = (x −3)(x + 3), instantly. 2. Perfect-square trinomials. What you get when you square a sum or difference: a2 + 2ab + b2 = (a + b)2 a2 −2ab + b2 = (a −b)2.

This time the middle terms add, giving the doubled 2ab. Recognizing this pattern is exactly what “completing the square” is about — the trick we’re about to use to crack every quadratic.

3. Sum and difference of cubes. Less famous, equally worth knowing: a3 −b3 = (a −b)(a2 + ab + b2), a3 + b3 = (a + b)(a2 −ab + b2).

4. The grand pattern: xn −1. This one holds for every whole-number power n: xn −1 = (x −1)  xn−1 + xn−2 + · · · + x + 1  .

In words: xn −1 always has (x −1) as a factor, and the other factor is every power of x from 0 up to n −1, all added together.

♦Why it’s true — skip on a first read, but the real brain-training Here’s the proof, and it uses a trick worth stealing. Multiply (x−1) through the long sum. The x hits every term, bumping each power up by one; the −1 copies each term unchanged with a minus sign: x · (xn−1 + · · · + x + 1) = xn + xn−1 + · · · + x2 + x −1 · (xn−1 + · · · + x + 1) = −xn−1 −· · · −x2 −x −1 Now add the two rows. Look down the columns: xn−1 cancels −xn−1, x2 cancels −x2, x cancels −x — every term in the middle annihilates its twin, like a row of dominoes knocking each other flat. This is a telescoping sum, because the whole thing collapses like a folding telescope. The only survivors are the very first term, xn, and the very last, −1, giving xn −1. ✓ And notice — difference of squares is just the n = 2 case of this very pattern. Rearrange it slightly — divide both sides by (x−1) — and it becomes the formula for a geometric sum: 1 + x + x2 + · · · + xn−1 = xn −1 x −1 .

Keep this one. You will meet it at least three more times: adding up any geometric chain (compound interest, a bouncing ball, a repeating decimal); probability (“how many tries until the first success?”); and complex numbers, where xn = 1 has n solutions arranged as the corners of a polygon on a circle — the roots of unity. Same algebra, four different worlds.

Polynomials, Factoring, and the Quadratic Formula

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