Problem 14 - College Algebra

Question

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

Step-by-Step Solution

Answer

The expansion of \((5a + 2)^3\) is \(125a^3 + 150a^2 + 60a + 8\).

Step 1: Identify the expression

We are given the expression \((5a + 2)^3\) to expand using the Binomial Theorem.

Step 2: Recall the Binomial Theorem

The Binomial Theorem states that \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). We will apply this theorem to expand the expression.

Step 3: Determine the values to substitute

In the binomial expression \((5a + 2)^3\), we have \(x = 5a\), \(y = 2\), and \(n = 3\).

Step 4: Calculate binomial coefficients

The binomial coefficients are given by \(\binom{n}{k}\). Compute these for \(n = 3\), which are \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\).

Step 5: Apply the Binomial Theorem

Using the coefficients, the expansion is: \((5a + 2)^3 = \binom{3}{0} (5a)^3 (2)^0 + \binom{3}{1} (5a)^2 (2)^1 + \binom{3}{2} (5a)^1 (2)^2 + \binom{3}{3} (5a)^0 (2)^3\).

Step 6: Simplify each term

- For \(k = 0\): \(1\cdot(5a)^3\cdot1 = 125a^3\)- For \(k = 1\): \(3\cdot(5a)^2\cdot2 = 150a^2\)- For \(k = 2\): \(3\cdot(5a)^1\cdot4 = 60a\)- For \(k = 3\): \(1\cdot1\cdot8 = 8\)

Step 7: Write the final expanded form

Combine all simplified terms to get the expanded form: \((5a + 2)^3 = 125a^3 + 150a^2 + 60a + 8\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a method used to express a mathematical expression made up of terms raised to powers. This concept is significant when working with binomials, which are algebraic expressions that consist of two terms connected by a plus or minus sign. For example, in the problem at hand, the binomial expression is

• \((5a + 2)\), raised to the power of 3.

To expand a polynomial using the Binomial Theorem, you need to replace the raised power with a series of terms that include coefficients, powers of each term in the binomial, and their products. This turns an expression like

• \((5a + 2)^3\)

into a polynomial composed of multiple terms after expansion. Each term of the expanded polynomial will have different powers of the binomial’s original terms. Understanding polynomial expansion is crucial in simplifying complex algebraic expressions.

Binomial Coefficients

Binomial coefficients are a vital part of expanding expressions using the Binomial Theorem. They are numbers that multiply the terms of the polynomial expansion and are often denoted as \(\binom{n}{k}\). These coefficients are found in Pascal’s Triangle, a useful tool allowing quick determination of these numbers.

In our example, the power \(n\) is 3, so we refer to the fourth row of Pascal's Triangle (remembering the top row is row zero). The binomial coefficients \(\binom{3}{k}\) for \(k = 0, 1, 2, 3\) are 1, 3, 3, and 1 respectively.

• These coefficients multiply the terms in the expansion according to their corresponding powers.

For instance, when expanding

• \((5a + 2)^3\)

we used these coefficients to form the expanded expression: \

• 125a^3, 150a^2, 60a, and 8.

The use of binomial coefficients is essential as they balance the powers and ensure an accurate representation of the expanded form.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition and multiplication. These expressions can represent real-world quantities or abstract mathematical ideas. Understanding how to manipulate these, especially through techniques like expansion, is a fundamental skill in algebra.

In algebra, when faced with expressions such as \((5a+2)^3\), we must expand them to simplify or reinterpret their components. The expanded expression from our example shows each individual term clearly, making the relationships between terms more apparent.

• Expressions like \(125a^3 + 150a^2 + 60a + 8\), obtained from expansion, are easier to integrate into larger algebraic solutions or equations.

Recognizing and working with algebraic expressions often involves combining like terms, factoring, and distributing, which are all steps that enhance a student's algebraic skills. These skills are foundational for higher-level mathematics and various applications in science and engineering.