Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (4 a-b)^{3} $$

Step-by-Step Solution

Answer

The expansion of \((4a - b)^3\) is \(64a^3 - 48a^2b + 12ab^2 - b^3\).

Step 1: Identify the Binomial Components

The binomial given is \((4a - b)\), and we need to expand its cube, \((4a - b)^3\), using the Binomial Theorem. Here, \(x = 4a\) and \(y = -b\), and the exponent \(n = 3\).

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 formula to expand \((4a - b)^3\), where \(x = 4a\), \(y = -b\), and \(n = 3\).

Step 3: Calculate the Binomial Coefficients

For \(n = 3\), the binomial coefficients are \(\binom{3}{0}\), \(\binom{3}{1}\), \(\binom{3}{2}\), and \(\binom{3}{3}\). These are 1, 3, 3, and 1, respectively.

Step 4: Expand Using Binomial Theorem

Using the calculated coefficients and the Binomial Theorem:- The term for \(k = 0\) is \(\binom{3}{0}(4a)^{3}( -b)^{0} = 1 \cdot (4a)^3 \cdot 1 = 64a^3\).- The term for \(k = 1\) is \(\binom{3}{1}(4a)^{2}(-b)^{1} = 3 \cdot 16a^2 \cdot (-b) = -48a^2 b\).- The term for \(k = 2\) is \(\binom{3}{2}(4a)^{1}(-b)^{2} = 3 \cdot 4a \cdot b^2 = 12a b^2\).- The term for \(k = 3\) is \(\binom{3}{3}(4a)^{0}(-b)^{3} = 1 \cdot 1 \cdot (-b)^3 = -b^3\).

Step 5: Combine the Terms

Combine all the terms obtained from the expansion: \(64a^3 - 48a^2 b + 12a b^2 - b^3\). Thus, the expansion of \((4a - b)^3\) is complete.

Key Concepts

Binomial Expansion

The concept of binomial expansion is a critical part of algebra that allows you to expand expressions that are raised to a power. The Binomial Theorem provides a formula to expand binomials of the form \((x + y)^n\) into a series of terms with powers and products of the variables x and y. This theorem is not only helpful in simplifying complex algebraic expressions but also in comprehending how polynomials work in various mathematical contexts.
The Binomial Theorem states that:

• Each term in the expansion has a binomial coefficient.
• The exponents of x decrease from n to 0, while those of y increase from 0 to n.
• Terms can be calculated using the formula \(\binom{n}{k} x^{n-k} y^k\), where k is the term number.

To apply this theorem, you identify the number of terms, calculate the binomial coefficients, and then compute each term in the series. It's particularly useful when exponentiating binomials quickly without the need for tedious multiplication. Applying this method to \((4a-b)^3\), you expand the binomial efficiently, leading to an expression like \(64a^3 - 48a^2b + 12ab^2 - b^3\). Understanding binomial expansion is essential for anyone delving into algebra and higher mathematics.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. These expressions form the building blocks of algebra and allow for the representation of formulas and problem-solving in math. In an expression like \((4a - b)^3\), we see a binomial expression that needs expanding.
Algebra helps us manage complexity:

• Variables like 'a' and 'b' stand in for unknown numbers, making generalization possible.
• Operations dictate how numbers and variables combine, such as addition, subtraction, multiplication, and exponentiation.
• Constants provide specific values within expressions, such as the numeric 4 in \(4a\).

Understanding how to manipulate and expand these expressions is crucial in algebra. Through the expansion, each term of the binomial might incorporate both coefficients and variables' powers. For instance, \((4a-b)^3\) results in varied terms, demonstrating how algebraic expressions represent diverse scenarios. Mastery of rather simple expressions like these leads to the ability to decode more complex ones and apply them in novel situations, from solving equations to modeling real-world phenomena.

Polynomial Expansion

Polynomial expansion refers to expressing a polynomial, particularly those in binomial form, in its fully expanded terms. A polynomial is an expression that can have multiple terms, and expansion helps us to rewrite such expressions in a simpler form for better understanding and computation.
This expansion is often facilitated by:

• Employing the Binomial Theorem, which provides a structured way to expand binomials like \((x + y)^n\).
• Utilizing algebraic identities to reduce the number of steps.
• Exploiting properties such as distributive law.

For our example \((4a - b)^3\), expansion results in a polynomial consisting of a sum of four terms, each being a product of powers of 'a' and 'b'. Each component in the expanded polynomial \(64a^3 - 48a^2b + 12ab^2 - b^3\) corresponds to a specific arrangement of the binomial multiplied out fully.
As one grows familiar with polynomial expansion, it becomes easier to analyze and work with mathematical expressions across different areas, including geometry, calculus, and even in real-world data approximation. Accurately expanding such expressions can simplify complex problems, allowing for more efficient solutions and insights.