I’m revising binomial expansion to strengthen my fundamentals, and I keep tripping over how to pick k and track the sign when there’s a negative term and a number in front of x. For example, I want the coefficient of x^2 in (3x − 2)^5.
My understanding is that the general term is C(5, k) (3x)^(5−k) (−2)^k with k starting at 0. So the power of x in that term is 5−k. To get x^2, I set 5−k = 2, so k = 3. Then I think the coefficient should come from C(5, 3) · 3^(5−3) · (−2)^3. This feels right, but I’m not fully confident I’m aligning k with the exponent correctly or handling the sign consistently.
Could someone please check whether this mapping from the target power of x to k is the right approach here? Also, is there a reliable step-by-step way to go from “target power of x” → “value of k” → “coefficient” for expressions like (ax + b)^n, especially when b is negative, without mixing up the sign or the power on a? If there’s a quick sanity check to catch sign errors or a missing factor of a, I’d love to learn it.
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3 Responses
You’ve lined everything up correctly. In (3x − 2)^5, the general term is C(5, k)(3x)^(5−k)(−2)^k, the x-power is 5 − k, so for x^2 you set 5 − k = 2, which gives k = 3. Then the coefficient is:
C(5, 3) · 3^(5−3) · (−2)^3 = 10 · 3^2 · (−8) = 10 · 9 · (−8) = −720.
A clean, reliable recipe for (ax + b)^n
– Target: coefficient of x^r.
– Step 1: r must be between 0 and n. If not, the coefficient is 0.
– Step 2: k = n − r. (k counts how many times you pick b; r counts how many times you pick ax.)
– Step 3: Coefficient = C(n, k) · a^r · b^k = C(n, n − r) · a^r · b^(n − r).
– Equivalently (often easier to remember): Coefficient = C(n, r) · a^r · b^(n − r). This uses C(n, r) = C(n, n − r).
– Step 4 (sign): The sign comes entirely from b^(n − r). If b < 0, the sign is negative exactly when n − r is odd. Why this is easy to keep straight - The power of a matches the power of x. Since a multiplies x, every time you choose an (ax), you get one x and one factor of a. So the coefficient must have a^r when the x-power is r. This avoids the common mix-up with a^(n − r). - The sign check is just parity of n − r if b is negative. Quick sanity checks I use - Parity sign check: If b < 0 and n − r is odd, the coefficient must be negative; if even, positive. - Exponent sum check: The powers on a and b should add to n: r + (n − r) = n. - Edge cases: r = 0 gives the constant term b^n; r = n gives the leading term a^n x^n. Your example revisited - Here, a = 3, b = −2, n = 5, r = 2. - k = n − r = 3. - Coefficient = C(5, 2) · 3^2 · (−2)^3 = 10 · 9 · (−8) = −720. - Sign check: n − r = 3 is odd and b < 0, so the coefficient should be negative. Matches. An alternate viewpoint (sometimes easier mentally) - Let u = ax. Then (ax + b)^n = (u + b)^n. - The coefficient of u^r is C(n, r) b^(n − r). - Substitute back u = ax to get the coefficient of x^r: C(n, r) a^r b^(n − r). - Same formula, but it spares you thinking about k at all. Tiny extra example for practice - Coefficient of x^4 in (2x − 5)^7: - r = 4 → coefficient = C(7, 4) · 2^4 · (−5)^(3) = 35 · 16 · (−125) = −70,000. - Sign check: n − r = 3 is odd and b < 0 ⇒ negative. Good. When I was learning this, I kept writing a^(n − r) out of habit because I was thinking in terms of “how many a’s are left,” which is wrong here. Switching to the substitution u = ax fixed it for me-once I remembered “the a travels with x,” the a^r factor became obvious, and my sign errors dropped when I did the quick parity check on n − r. For a clear refresher on the binomial theorem and finding specific terms, this Khan Academy page is solid: https://www.khanacademy.org/math/algebra/polynomials/polynomial-binomial-theorem/a/binomial-theorem
You’ve got the right compass: in (ax + b)^n, the x^p term comes from k = n − p, and its coefficient is C(n, k) a^{n−k} b^k-so the sign is baked into b^k (negative if b < 0 and k is odd). For (3x − 2)^5 and x^2, take k = 5 − 2 = 3, giving coefficient C(5,3)·3^2·(−2)^3 = 10·9·(−8) = −720; sanity check: k is odd ⇒ negative-nice.
Spot on-think of k as how many times you pick the constant: in (ax + b)^n choose k so n−k equals your target x-power, then the coefficient is C(n, k) a^k b^{n−k}, and with b negative the sign flips whenever k is odd.
Example: 5−k=2 ⇒ k=3, so coeff = C(5,3)·3^{5−3}·(−2)^3 = 10·9·(−8) = −720 (quick check: +,−,+,−,+,− pattern, so the x^2 term is negative).