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3 Responses
Think of (2x − 5)^7 as seven little factors sitting in a row, and to get an x^3 you “pick” the x from exactly three of them and the constant from the other four-like choosing three red socks from a drawer and leaving four blue behind. So the coefficient comes from: choose which 3 spots give you x (that’s C(7,3)), then each chosen spot contributes a 2 from 2x (so 2^3), and each of the remaining four contributes a −5 (so (−5)^4). I almost wrote (−5)^3 at first-brain did a hop-but we need four constants because only three factors give x, so it’s (−5)^4. Since the 4th power makes it positive, no sign flip in the end.
Crunching it: C(7,3) = 35, 2^3 = 8, and (−5)^4 = 625. Multiply: 35 × 8 = 280, and 280 × 625 = 175000. So the coefficient of x^3 in (2x − 5)^7 is 175000. Hope this helps!
Think of (2x−5)^7 as a seven-scoop sundae where each scoop is either “2x-flavor” or “−5-flavor.” To get an x^3 term, you must pick the 2x scoop exactly three times (for x·x·x) and the −5 scoop the other four times-so it’s a choose-not-order situation: choose 3 of the 7 spots for 2x. That gives the coefficient as C(7,3)·(2)^3·(−5)^4. I almost tripped over the sign (since there’s a minus), but the −5 appears four times, and four is even, so it ends up positive. Compute: C(7,3)=35, 2^3=8, (−5)^4=625, so the x^3 coefficient is 35·8·625 = 175000. Tiny warm-up example: in (3x+1)^4, the x^2 coefficient comes from choosing two 3x’s and two 1’s, so it’s C(4,2)·3^2·1^2 = 6·9 = 54. I briefly wondered if permutations mattered (they don’t here-order doesn’t change the product), so combinations are the right sock-drawer rule!
Yes-use the binomial “pick-and-mix” rule: in (ax + b)^n, the x^k term comes from choosing k of the n factors to supply ax and the rest to supply b, so its coefficient is C(n,k)·a^k·b^(n−k). Here, (2x − 5)^7: to get x^3, choose 3 of the 7 factors to contribute 2x and the other 4 to contribute −5, giving coefficient C(7,3)·2^3·(−5)^4 = 35·8·625 = 175000 (positive, since the power of −5 is even). Quick check with a simpler example: in (3x + 1)^5, the x^2 coefficient is C(5,2)·3^2·1^3 = 10·9 = 90.