Problem 64 - College Algebra

Question

Explain how to use the Binomial Theorem to expand a binomial. Provide an example with your explanation.

Step-by-Step Solution

Answer

The Binomial Theorem can be used to expand a binomial expression \((x + y)^3\) into \(x^3 + 3x^2y + 3xy^2 + y^3\).

Step 1: Understand the Binomial Theorem

We use the Binomial Theorem to expand binomial expressions. The general formula is \((a + b)^n = \sum_{k=0}^{n} {n \choose k}a^{n-k}b^k\), where \({n \choose k}\) are the binomial coefficients. These can be computed using the formula \({n \choose k} = \frac{n!}{k!(n - k)!}\). '!' denotes factorial, defined as the product of an integer and all integers below it down to 1.

Step 2: Apply the Binomial Theorem to an Example

Let's choose a binomial expression \((x + y)^3\). According to the theorem, \((x + y)^3 = \sum_{k=0}^{3} {3 \choose k}x^{3-k}y^k\). This sum has four terms, corresponding to \(k = 0, 1, 2, 3\). Calculate each term separately: {3 \choose 0}x^3y^0, {3 \choose 1}x^2y^1, {3 \choose 2}x^1y^2, and {3 \choose 3}x^0y^3.

Step 3: Simplify the Result

By calculating binomial coefficients and simplifying each term, the expansion becomes \(x^3 + 3x^2y + 3xy^2 + y^3\).