### 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 *ax*^{b}*y*^{c}, 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 —

**Examples** :

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*x*^{0}= 1, often unwritten); - the powers of
*y*start at 0 and increase by 1 until they reach*n*; - the
*n*th 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 2
^{n}; and - there will be
*n*+ 1 terms in the expression after combining like terms in the expansion.

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