Problem 31 - Algebra 2

Question

Expand each binomial. $$ (x+1)^{6} $$

Step-by-Step Solution

Answer

\((x + 1)^6 = x^{6} + 6x^{5} + 15x^{4} + 20x^{3} + 15x^{2} + 6x + 1\).

Step 1: Applying the Binomial Theorem

We can start by applying the binomial theorem for our problem. The equation we use from the binomial theorem is also known as the binomial expansion: \((x + 1)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} 1^{k}\). Considering 1 raised to any power is 1, we can simplify this equation to: \((x + 1)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k}\).

Step 2: Performing the Expansion

Next, we substitute values of \(k\) from 0 to 6 into the equation. The coefficients \(\binom{6}{k}\) are computed by 6 Choose k (also known as Combination). The simplified expansion is: \((x + 1)^6 = \binom{6}{0}x^{6} + \binom{6}{1}x^{5} + \binom{6}{2}x^{4} + \binom{6}{3}x^{3} + \binom{6}{4}x^{2} + \binom{6}{5}x^{1} + \binom{6}{6}x^{0}\).

Step 3: Simplification of the Expansion

In this step, we simply have to compute the binomial coefficients and rewrite the equation. \((x + 1)^6 = x^{6} + 6x^{5} + 15x^{4} + 20x^{3} + 15x^{2} + 6x + 1\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that provides a method to expand expressions raised to any given power. When you see expressions like \((x + 1)^n\), the Binomial Theorem allows you to express this expanded form efficiently. Essentially, it uses the formula: \[(x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^{k}\] where \(n\) is any positive integer, and \(\binom{n}{k}\) are binomial coefficients.

In our original exercise, exploring \((x + 1)^6\) involves applying this theorem with \(a = 1\) and \(n = 6\). Each term in this expansion corresponds to a value of \(k\) from 0 to \(n\), showing a blended mix of powers of \(x\) and the constants.

Binomial Coefficients

Binomial coefficients are integral to working through binomial expansions. They arise from the expression \(\binom{n}{k}\), which is read as "n choose k" and represents combinations in mathematics. This counts the number of ways to choose \(k\) elements from \(n\) elements without considering the order.

For the expansion of \((x + 1)^6\), these coefficients, \(\binom{6}{0}, \binom{6}{1}, \binom{6}{2}, \ldots, \binom{6}{6}\), are easily computed by specific formulas or Pascal’s Triangle. Each coefficient corresponds to a unique term in the expression, influencing its associated power of \(x\). The computed coefficients are 1, 6, 15, 20, 15, 6, and 1, respectively.

Algebraic Expressions

Algebraic expressions refer to combinations of variables, numbers, and operators in math. In the expansion of a binomial using the Binomial Theorem, we break down complex expressions into simpler parts.

For example, when you expand \((x + 1)^6\), it becomes a lengthy expression comprising different powers of \(x\), each with specific coefficients. This expanded form is made of terms like \(x^6\), \(6x^5\), and so on, each contributing to a complete representation of the original problem.

Understanding algebraic expressions and their proper handling allows for a straightforward simplification process, aiding you in solving wider mathematical problems.