Find the coefficient of x4 in the expansion of (3 − 2x)6.
- General term: 6Cr × 36 − r × (−2x)r. For x4 take r = 4.
- 6C4 = (6 × 5)/(2 × 1) = 15; 32 = 9; (−2)4 = 16.
- Coefficient = 15 × 9 × 16 = 135 × 16.
Answer: 2160
The binomial theorem
Maths: Analysis and Approaches · Number and algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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The idea
The binomial theorem gives the expansion of (a + b)n without multiplying out the brackets one at a time. The expansion has n + 1 terms. In each term the powers of a and b add up to n: the power of a falls from n to 0 while the power of b rises from 0 to n.
The number in front of each term is a binomial coefficient, nCr. You can read it from row n of Pascal's triangle or work it out with factorials.
Most questions ask for one term only. Use the general term and choose the value of r that gives the power you need. Keep the whole of b, with its sign and its number, inside the bracket.
Rules to remember
- (a + b)n = an + nC1 an − 1b + nC2 an − 2b2 + ... + bn
- General term: nCr × an − r × br, for r = 0, 1, ..., n
- nCr = n!/(r! × (n − r)!); nC0 = nCn = 1; nCr = nCn − r
- Pascal's triangle: row 4 is 1, 4, 6, 4, 1; row 5 is 1, 5, 10, 10, 5, 1
- The expansion of (a + b)n has n + 1 terms
- If b is negative the signs alternate: (1 − x)3 = 1 − 3x + 3x2 − x3
- Term independent of x: choose r so that the powers of x cancel to x0
Common mistake
Raising only the x to the power and forgetting the number and sign with it: in (3 − 2x)6 the x4 term needs (−2x)4 = 16x4, not just x4.
Worked example
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