I’m trying to grab the coefficient of x^3 in (2x − 1/x)^8 without writing out all nine terms, because that’s a pain. I know the general term is C(8, r)(2x)^{8−r}(−1/x)^r, but I keep messing up which r actually lands me on x^3. I tried setting up something like 8 − 2r = 3 for the power of x, but I’m not confident I’m handling the minus signs and the 1/x properly, and then I lose track of the overall sign and coefficient once I pick r. Is there a clean, no-nonsense way to pick r (and the sign) quickly, maybe a simple rule I can remember? And if I wanted the constant term instead, is it the same trick?
I also tried doodling Pascal’s triangle to see patterns, but that didn’t help with the x-powers, so maybe that’s a dead end. Any help appreciated!
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3 Responses
Think of (2x − 1/x)^8 like walking down a staircase two steps at a time: every time you choose a −1/x instead of a 2x, the power of x drops by 2. Concretely, the general term is C(8, r)(2x)^(8−r)(−1/x)^r, whose x-power is 8 − 2r, and the sign is just (−1)^r. That gives a quick rule: if you want x^k, solve 8 − 2r = k, i.e., r = (8 − k)/2. If r isn’t an integer, there’s no such term (because we only land on even exponents: 8, 6, 4, 2, 0, −2, …). So for x^3, r = (8 − 3)/2 = 2.5, not an integer-boom, the coefficient is 0. For the constant term (k = 0), r = (8 − 0)/2 = 4 works, and the coefficient is C(8,4)·2^(8−4)·(−1)^4 = 70·16·1 = 1120. Quick memory hook: target exponent must match the parity of 8 (even), r = (8 − k)/2 picks the term, and the sign is positive if r is even, negative if r is odd.
Quick parity check: every term is x^(8−2r), so only even powers appear-x^3 never shows (coefficient 0); the constant term comes from 8−2r=0 → r=4, giving C(8,4)·2^4 with a plus sign = 1120.
Want to try the same trick on, say, (3x + 1/x)^7 to see what happens?
Write the general term as C(8, r)(2x)^(8−r)(−1/x)^r. The power of x in that term is (8−r) − r = 8 − 2r, so to hit x^3 you’d need 8 − 2r = 3, which gives r = 2.5. Since r must be an integer, there’s no such term, so the coefficient of x^3 is 0. A quick rule to remember: when one factor has x^1 and the other has x^−1, exponents in the expansion jump by 2, so only even powers of x appear. The sign on the rth term is (−1)^r. For the constant term, use the same setup: 8 − 2r = 0 gives r = 4, so its coefficient is C(8, 4)·2^4 with a positive sign (since r is odd), i.e., 1120.