Problem 13 - College Algebra

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (5 x-1)^{3} $$

Step-by-Step Solution

Answer

The expansion of \((5x-1)^3\) using the Binomial Theorem is \(125x^3 - 75x^2 + 15x - 1\).

Step 1: Understanding the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \left(\begin{array}{l}n\k\end{array}\right) a^{n-k} b^k\), where \(\left(\begin{array}{l}n\k\end{array}\right)\) denotes the binomial coefficient 'n choose k'. This indicates that each term in the expansion is formed by multiplying a term from the binomial by a binomial coefficient.

Step 2: Apply the Binomial Theorem to the given binomial

We rewrite \((5x-1)^3\) using the Binomial Theorem: \[(5x-1)^3 = \sum_{k=0}^{3} \left(\begin{array}{l}3\k\end{array}\right) (5x)^{3-k} (-1)^k\]

Step 3: Calculate each term in the binomial expansion

Compute each term in the series:\[\left(\begin{array}{l}3\0\end{array}\right) (5x)^3 (-1)^0 + \left(\begin{array}{l}3\1\end{array}\right) (5x)^2 (-1)^1 + \left(\begin{array}{l}3\2\end{array}\right) (5x)^1 (-1)^2 + \left(\begin{array}{l}3\3\end{array}\right) (5x)^0 (-1)^3\]

Step 4: Final simplification of the binomial expansion

Simplify each term to get:\[125x^3 - 75x^2 + 15x - 1\]

Key Concepts

Binomial Expansion

The Binomial Theorem provides a powerful way to expand expressions that are raised to a power in the form ewline \((a+b)^n\). Binomial expansion involves expressing the binomial raised to a power as the sum of terms known as binomial coefficients multiplied by the individual terms of the binomial raised to varying powers. Each term in the binomial expansion follows a pattern based on the powers of the binomial's terms and the index of the term in the series.

When using the Binomial Theorem for expansion, the general form is \(\sum_{k=0}^{n} \left(\begin{array}{l}n\k\end{array}\right) a^{n-k} b^k\), where \(a\) and \(b\) are the terms from the binomial and \(n\) is the exponent. The expansion of \((a+b)^n\) will result in \(n+1\) terms, where the value at each position is obtained by choosing \(k\) elements from \(n\) possibilities, giving us the binomial coefficients.
For instance, expanding \((5x-1)^3\) means we would apply the theorem and get a series of terms in the form of \(\text{coefficient} \times (5x)^{3-k} \times (-1)^k\), leading us to a final expanded algebraic expression.

Binomial Coefficient

A critical aspect of binomial expansion is understanding binomial coefficients. The binomial coefficient, denoted as \(\left(\begin{array}{l}n\k\end{array}\right)\), represents the number of ways to choose \(k\) elements out of a set of \(n\) elements, and it's also known as 'n choose k'. It is calculated using the formula \(\left(\begin{array}{l}n\k\end{array}\right) = \frac{n!}{k!(n-k)!}\), where \(n!\) denotes the factorial of \(n\), which is the product of all positive integers up to \(n\).

For simplification in binomial expansions, understanding how to compute these coefficients is essential, as they directly affect the multipliers of each term in the series. For example, the coefficients for the expansion of \((5x-1)^3\) will be calculated for \(k=0\) to \(k=3\), providing us the necessary multipliers for each term of our binomial expansion.

Algebraic Expression Simplification

After computing the individual terms in the binomial expansion, which includes determining the binomial coefficients, the next step is to simplify the algebraic expression. This means performing the arithmetic operations to consolidate like terms and arriving at the simplest form of the expression.

Simplification involves careful arithmetic work such as multiplication of the coefficients by the corresponding powers of the binomial terms, handling of signs (especially for negative terms), and combining like terms, which are terms with the same variables raised to the same power. For instance, in the expression \((5x-1)^3\), after applying the Binomial Theorem and finding each term, we sum them up to get the simplified form \(125x^3 - 75x^2 + 15x - 1\). This final step is critical, as it provides the expanded expression in its most concise and useful form, whether for further algebraic manipulation or for practical application in problems.