Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (5 a+2)^{3} $$

Step-by-Step Solution

Answer

The expanded form is \( 125a^3 + 150a^2 + 60a + 8 \).

Step 1: Understand the Binomial Theorem

The Binomial Theorem provides a way to expand any binomial expression \( (x + y)^n \) into a sum involving terms of the form \( \binom{n}{k}x^{n-k}y^k \) where \( \binom{n}{k} \) is the binomial coefficient, \( n \) is the power, and \( k \) is the index of each term.

Step 2: Identify the Components

In the expression \( (5a + 2)^3 \), we can identify \( x = 5a \), \( y = 2 \), and \( n = 3 \). We will use these to expand the expression using the Binomial Theorem.

Step 3: Calculate the Binomial Coefficients

The binomial coefficients for \( (x + y)^3 \) are \( \binom{3}{0} = 1 \), \( \binom{3}{1} = 3 \), \( \binom{3}{2} = 3 \), and \( \binom{3}{3} = 1 \). These coefficients will be used to multiply each respective term in the expansion.

Step 4: Expand the Binomial Expression

The expansion of \( (5a + 2)^3 \) is computed as follows:- \( \binom{3}{0}(5a)^3(2)^0 = 1 \times (5a)^3 = 125a^3 \)- \( \binom{3}{1}(5a)^2(2)^1 = 3 \times 25a^2 \times 2 = 150a^2 \)- \( \binom{3}{2}(5a)^1(2)^2 = 3 \times 5a \times 4 = 60a \)- \( \binom{3}{3}(5a)^0(2)^3 = 1 \times 8 = 8 \)

Step 5: Combine the Terms

Add the terms from the expanded expression: \( 125a^3 + 150a^2 + 60a + 8 \). This is the fully expanded form of the binomial \( (5a + 2)^3 \).

Key Concepts

Binomial Expansion

The concept of binomial expansion refers to the process of expanding an expression raised to a power. For the expression \((x + y)^n\), the expansion involves multiple terms, each with a specific coefficient and power of \(x\) and \(y\). These coefficients are determined using the Binomial Theorem, which offers a systematic approach to finding each term of the expansion. The expansion essentially takes a compact binomial and expresses it as a sum of individual terms, each term involving multiplication of powers.

To understand binomial expansion better, imagine a simple expression such as \((a+b)^2\). Expanding this step-by-step means calculating each term separately:

• The first term, formed by squaring \(a\), results in \(a^2\).
• The second term comes from calculating \(2ab\), which is a result of multiplying \(a\) and \(b\), and then multiplying by the binomial coefficient 2.
• The final term is \(b^2\), which is obtained by squaring \(b\).

Understanding this process with small powers helps in visualizing and performing binomial expansions for larger powers in a similar manner.

Binomial Coefficients

Binomial coefficients are crucial elements in a binomial expansion. They are the numerical factors that precede each term in the expansion, derived from the binomial formula. The notation for binomial coefficients is \(\binom{n}{k}\), which is read as "n choose k," signifying the number of ways to choose \(k\) objects from \(n\) total objects.

The value of a specific binomial coefficient \(\binom{n}{k}\) can be computed using the formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
This formula calculates the coefficient by taking the factorial of the number of terms \(n\), divided by the factorial of the term's position \(k\) and the factorial of the difference \(n-k\). These coefficients are employed to balance the contributions of each pair of terms \(x^{n-k}\) and \(y^k\) in the binomial expansion.

A typical pattern occurs in binomial coefficients, resembling the entries found in the Pascal’s Triangle, which offers a handy visualization of how coefficients are structured.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables (like \(a\) or \(b\)), and operational symbols (like +, -, *, and /). In the context of binomial expansions, algebraic expressions often appear in the form of binomials, which are expressions containing the sum of two terms, such as \((x + y)\).

Algebraic expressions are pivotal in understanding and performing expansions, as they form the basis of our work with the Binomial Theorem. Each term in the expansion is an algebraic expression itself, involving constants and variables. For example, when expanding \((5a + 2)^3\), each resulting term like \(125a^3\) or \(8\), is a separate algebraic expression involving coefficients, powers of \(a\), and constants.

Operations on algebraic expressions often require clear steps and attention to the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This simplicity in operations holds significant importance when solving problems involving complex binomials.