Expanding a power of a binomial
The binomial theorem gives a quick way to expand (a + b)n without multiplying the brackets out one at a time. The coefficients are the combinations nCr, which also form Pascal's triangle.
Key idea
(a + b)n = nC0an + nC1an−1b + nC2an−2b2 + … + nCnbn. There are n + 1 terms.
The general term is nCr an−r br. As r increases by 1 the power of a drops by 1 and the power of b rises by 1, while the two powers always add up to n. To find one particular term, choose the r that gives the power you want.
Worked example
Find the coefficient of x3 in the expansion of (2 + x)5.
The general term is 5Cr(2)5−rxr. For x3, r = 3: 5C3(2)2x3 = 10 × 4 × x3 = 40x3. The coefficient is 40.
Remember
- (a + b)n has n + 1 terms.
- Substituting x = 1 into (1 + x)n shows the coefficients add to 2n.
- The powers of a and b in each term always sum to n.