Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand and simplify the expression. \(\left(2 x^{3}-y\right)^{5}\)

Step-by-Step Solution

Answer

The expanded and simplified form of \(\left(2 x^{3}-y\right)^{5}\) is \(32x^{15} - 160x^{12}y + 320x^{9}y^{2} - 320x^{6}y^{3} + 160x^{3}y^{4} - y^{5}\).

Step 1: Identify a, b, and n

In our given expression, \(a = 2x^{3}\), \(b = -y\), and \(n = 5\)

Step 2: Use the Binomial Theorem

Using the Binomial Theorem, we write \((2x^{3} - y)^{5}\) as the summation \(\sum_{k=0}^{5}\left( \begin{matrix} 5 \ k \end{matrix} \right)(2x^{3})^{5-k}(-y)^{k}\)

Step 3: Expand the series

On expanding the above series, we get \((2x^{3})^{5} - 5(2x^{3})^{4}(-y) + 10(2x^{3})^{3}(-y)^2 - 10(2x^{3})^{2}(-y)^3 + 5(2x^{3})(-y)^4 - (-y)^5\).

Step 4: Simplify the expression

Finally, simplify each term in the expression. This results in \(32x^{15} - 160x^{12}y + 320x^{9}y^{2} - 320x^{6}y^{3} + 160x^{3}y^{4} - y^{5}\).

Key Concepts

Binomial Expansion Step by Step

Understanding binomial expansion can be much easier when we break it down into detailed steps. Using the Binomial Theorem, we can expand a binomial expression that is raised to a power without multiplying it out the long way. Let's walk through the process step by step, using the example from the exercise, \( (2x^3 - y)^5 \).

Step 1: Identify the components of the binomial, which we'll denote as \(a\) and \(b\), and the exponent, denoted as \(n\). For our example, \(a = 2x^3\), \(b = -y\), and \(n = 5\).
Step 2: Apply the Binomial Theorem, which states that \( (a + b)^n = \sum_{k=0}^{n} \left( \begin{matrix} n \ k \end{matrix} \right)(a)^{n-k}(b)^{k} \), where \( \left( \begin{matrix} n \ k \end{matrix} \right) \) represents the binomial coefficient, corresponding to the number of ways to choose \(k\) elements from a set of \(n\).
Step 3: Expand the series by calculating each term. This is a systematic process of multiplying \( a \) and \( b \) raised to their respective powers and then scaling by the binomial coefficient.
Step 4: The final step is to simplify the expression. Multiply out the coefficients with the powers of \( a \) and \( b \) and add or subtract as needed. After simplification, we get \(32x^{15} - 160x^{12}y + 320x^{9}y^{2} - 320x^{6}y^{3} + 160x^{3}y^{4} - y^{5}\).

By following these steps, binomial expansion is delineated into a clear, approachable process.

Simplifying Algebraic Expressions

The concept of simplifying algebraic expressions is vital when working with polynomial expansions such as binomial expansions. Simplification involves reducing an expression to its simplest form by performing operations like multiplication and addition, and combining like terms. In our given problem, after expanding the binomial, we need to simplify the polynomial terms to make the expression more understandable and usable.

When simplifying, pay close attention to the exponents and coefficients. For instance, when you take \( (2x^3)^5 \), you are raising both the coefficient (2) and the variable (\(x^3\)) to the 5th power. This gives us \( 32x^{15} \), using the property of exponents that states multiplying powers with the same base requires adding the exponents. Similar steps are taken for each term in the expanded binomial. Note that simplifying also requires careful handling of negative signs, especially when dealing with alternating terms as seen in binomial expansions.

Ultimately, simplification is about making expressions as concise as possible, which often makes subsequent calculations or understanding the properties of the equation much easier.

Polynomial Expansion

Polynomial expansion is the process of taking a polynomial expression that has been raised to a power and expanding it into a sum of terms. Binomial expansion is a specific case of polynomial expansion where the polynomial has two terms. The process involves distributing the power over the terms of the polynomial and applying combinatorial math to determine the coefficients.

For a binomial, the expansion follows the pattern given by the Binomial Theorem. For a polynomial with more than two terms, the process can become more complex and may involve using the Multinomial Theorem or applying the binomial expansion repeatedly.

Polynomial expansions play a crucial role in various areas of mathematics, including calculus and algebra. They are used to simplify complex calculations involving powers of polynomials and can be found in applications ranging from physics to finance.