Binomial Theorem 101 Sample Questions
Binomial Theorem 101 Sample Questions | Higher Mathematics Questions
The binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are non-negative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example (for n = 4),
According to the theorem,
it is possible to expand any non-negative power of (x + y) into a sum of the form —
Several patterns can be observed from these examples. In general, for the expansion (x + y)n:
- the powers of x start at n and decrease by 1 in each term until they reach 0 (with x0 = 1, often unwritten);
- the powers of y start at 0 and increase by 1 until they reach n;
- the nth row of Pascal’s Triangle will be the coefficients of the expanded binomial when the terms are arranged in this way;
- the number of terms in the expansion before like terms are combined is the sum of the coefficients and is equal to 2n; and
- there will be n + 1 terms in the expression after combining like terms in the expansion.
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