Maths · Mathematical induction and binomial theorem. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The binomial theorem gives the expansion of (a + x)n for a positive integer n without multiplying out the brackets. The expansion has n + 1 terms. In each term the powers of a and x add up to n: the power of a falls from n to 0 while the power of x rises from 0 to n.
The numbers in front, nC0, nC1, ..., nCn, are the binomial coefficients. They are equal at equal distances from the two ends.
You rarely need the whole expansion. The general term Tr+1 gives any single term directly: a named term, the term with a given power of x, the term independent of x, or the middle term.
Students use r = 4 for the 4th term, or raise only x to the power when the term is 2x or −x. The term Tr+1 has r one less than the term number, and the whole of the second quantity, with its sign and coefficient, is raised to the power r.
Answer: The term independent of x is the 3rd term, 240.
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