Problem 13 - College Algebra

Question

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

Step-by-Step Solution

Answer

The expanded form is \( 64a^3 - 48a^2b + 12ab^2 - b^3 \).

Step 1: Identify the Binomial and Exponent

The expression given is \( (4a - b)^3 \). Here, the binomial is \((4a - b)\) and the exponent is \(3\). The Binomial Theorem states that \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\).

Step 2: Write the Binomial Expansion Formula

According to the Binomial Theorem, the expansion of \((x+y)^n\) can be written as:\[ (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]In this problem, identify \(x = 4a\), \(y = -b\), and \(n = 3\).

Step 3: Calculate Binomial Coefficients

Calculate the binomial coefficients \(\binom{3}{k}\) for \(k = 0, 1, 2, 3\).- \(\binom{3}{0} = 1\)- \(\binom{3}{1} = 3\)- \(\binom{3}{2} = 3\)- \(\binom{3}{3} = 1\)

Step 4: Apply the Binomial Theorem

Substitute values into the expansion formula:\[ (4a-b)^3 = \binom{3}{0} (4a)^3 (-b)^0 + \binom{3}{1} (4a)^2 (-b)^1 + \binom{3}{2} (4a)^1 (-b)^2 + \binom{3}{3} (4a)^0 (-b)^3\]

Step 5: Simplify Each Term

1. \(\binom{3}{0} (4a)^3 (-b)^0 = 1 \cdot 64a^3 \cdot 1 = 64a^3\) 2. \(\binom{3}{1} (4a)^2 (-b)^1 = 3 \cdot 16a^2 \cdot (-b) = -48a^2b\) 3. \(\binom{3}{2} (4a)^1 (-b)^2 = 3 \cdot 4a \cdot b^2 = 12ab^2\) 4. \(\binom{3}{3} (4a)^0 (-b)^3 = 1 \cdot 1 \cdot (-b)^3 = -b^3\)

Step 6: Combine All Terms

Combine the results from each term to form the expanded expression:\[ 64a^3 - 48a^2b + 12ab^2 - b^3 \]

Key Concepts

Binomial Expansion

The concept of binomial expansion is crucial when dealing with expressions of the form \((x+y)^n\). This is where the Binomial Theorem comes into play. The theorem allows us to expand a binomial raised to a power by summing multiple terms. Each term is a product that includes a specific binomial coefficient, a power of the first term in the binomial, and a power of the second term. This method provides a more straightforward way to expand and simplify expressions without manually multiplying them repeatedly.

The expansion is determined by the formula:

• \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\)

Each term in this summation represents one part of the broader expression. For example, the binomial expansion of \((4a-b)^3\) results in several terms, each calculated using the respective binomial coefficient and powers of \(4a\) and \(-b\). Understanding this framework simplifies working with complex polynomials by breaking them down into more manageable parts.

Binomial Coefficients

Binomial coefficients play a pivotal role in the binomial expansion. They are the numbers that appear as coefficients in the expansion of a binomial raised to a power. The notation \(\binom{n}{k}\) represents the binomial coefficient, which can be calculated as:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

These coefficients reflect the number of possible combinations of \(k\) items chosen from \(n\) items without regard to order.

For the expression \((4a-b)^3\), the binomial coefficients are:

• \(\binom{3}{0} = 1\)
• \(\binom{3}{1} = 3\)
• \(\binom{3}{2} = 3\)
• \(\binom{3}{3} = 1\)

Each of these coefficients is used to scale the terms of the expansion. Knowing how to calculate binomial coefficients correctly is essential for accurately applying the expansion formula and simplifying expressions like \((4a-b)^3\).

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients. They involve operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. With polynomials, operations are carried out over terms that follow \(ax^n + bx^{n-1} + \ldots + zx^0\) where each term is a power of \(x\) multiplied by a coefficient.

In the exercise, the binomial \((4a-b)^3\) is essentially a form of a polynomial expansion. The result of expanding this binomial is another polynomial which features distinct terms:

• \(64a^3\)
• \(-48a^2b\)
• \(12ab^2\)
• \(-b^3\)

The power of polynomials lies in their ability to represent complex relationships simply. By understanding how to manipulate and expand these expressions using the Binomial Theorem and binomial coefficients, we can break down more formidable mathematical challenges into easy, calculable steps.