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Binomial Theorem 101 Sample Questions

Binomial Theorem 101 Sample Questions | Higher Mathematics Questions

Binomial Theorem 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 —

(x+y)^{n}={n \choose 0}x^{n}y^{0}+{n \choose 1}x^{n-1}y^{1}+{n \choose 2}x^{n-2}y^{2}+\cdots +{n \choose n-1}x^{1}y^{n-1}+{n \choose n}x^{0}y^{n},

Examples :

{\displaystyle {\begin{aligned}(x+y)^{3}&=x^{3}+3x^{2}y+3xy^{2}+y^{3},\\[8pt](x+y)^{4}&=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4},\\[8pt](x+y)^{5}&=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5},\\[8pt](x+y)^{6}&=x^{6}+6x^{5}y+15x^{4}y^{2}+20x^{3}y^{3}+15x^{2}y^{4}+6xy^{5}+y^{6},\\[8pt](x+y)^{7}&=x^{7}+7x^{6}y+21x^{5}y^{2}+35x^{4}y^{3}+35x^{3}y^{4}+21x^{2}y^{5}+7xy^{6}+y^{7}.\end{aligned}}}

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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