Maths WorkshopMaths problems with full solutions
Exercise 402Combinatorics › Binomial coefficientsHigh schoolAdded 3 Nov 2026

A specific expansion

#expansion#binomial theorem#computation

Statement

Expand (2x−3)4(2x-3)^4.

Hint

Apply binomial theorem with a=2xa=2x, b=−3b=-3.

Show solutionHide solution

Solution

Formula. (2x−3)4=∑k=04(4k)(2x)4−k(−3)k(2x-3)^4=\sum_{k=0}^{4}\binom{4}{k}(2x)^{4-k}(-3)^k.

Expand.
- k=0k=0: (40)(2x)4=16x4\binom{4}{0}(2x)^4=16x^4
- k=1k=1: (41)(2x)3(−3)=−96x3\binom{4}{1}(2x)^3(-3)=-96x^3
- k=2k=2: (42)(2x)2(9)=216x2\binom{4}{2}(2x)^2(9)=216x^2
- k=3k=3: (43)(2x)(−27)=−216x\binom{4}{3}(2x)(-27)=-216x
- k=4k=4: (44)(81)=81\binom{4}{4}(81)=81

Conclusion. (2x−3)4=16x4−96x3+216x2−216x+81(2x-3)^4=16x^4-96x^3+216x^2-216x+81.