Problem 13 - College Algebra

Question

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(5 x-1)^{3}$$

Step-by-Step Solution

Answer

The expanded form of the binomial (5x-1)^3 is 125x^3 - 75x^2 + 15x - 1.

Step 1: Identify the variables

In this case, a=(5x), b=(-1) and n=3.

Step 2: Apply binomial theorem

The binomial theorem states that (a+b)^n = \(\sum_{k=0}^{n}\) (n choose k) * a^(n-k) * b^k. For (5x-1)^3, it will be expanded to \(\sum_{k=0}^{3}\) (3 choose k) * (5x)^(3-k) * (-1)^k.

Step 3: Expansion

Now, we'll compute each term of the summation: \nTerm 1: (3 choose 0) * (5x)^3 * (-1)^0 = 1*125x^3*1 = 125x^3\nTerm 2: (3 choose 1) * (5x)^2 * (-1)^1 = 3*25x^2*(-1) = -75x^2\nTerm 3: (3 choose 2) * (5x) * (-1)^2 = 3*5x*1 = 15x\nTerm 4: (3 choose 3) * (-1)^3 = 1*(-1)=-1

Step 4: Combine all terms for final solution

Combine all terms: 125x^3 - 75x^2 + 15x - 1

Key Concepts

Polynomial Expansion

The concept of polynomial expansion revolves around the idea of expressing a power of a binomial as the sum of terms. The Binomial Theorem provides a straightforward method for doing this. For any binomial \((a + b)^n\), it allows us to expand it into a series of terms derived from the coefficients known as binomial coefficients.

These coefficients are found using "n choose k," represented as \(\binom{n}{k}\). It tells us how many ways we can choose \(k\) items from \(n\) without regard to the order. For example, expanding \((5x - 1)^3\) shows how these coefficients combine with powers of each component \((5x)\) and \((-1)\) to form the expanded polynomial:

• Term 1: \((5x)^3\)
• Term 2: \(3(5x)^2(-1)\)
• Term 3: \(3(5x)(-1)^2\)
• Term 4: \((-1)^3\)

By multiplying these components, we construct the whole polynomial as \(125x^3 - 75x^2 + 15x - 1\). This method helps us simplify complex expressions and better understand their structure.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. In the case of polynomial expansion using the Binomial Theorem, we deal with algebraic expressions such as \((5x-1)^3\).

Each algebraic expression can be expanded into simpler terms through systematic operations. Here's how it looks once it's expanded:

• The operation involves computing the power of each term, and multiplying by coefficients.
• The terms resulting from the expansion are collected and simplified to form the final expression.

In our example, the components \((5x)\) and \((-1)\) are raised to powers, then multiplied by coefficient values. The outcome is a systematic arrangement:

• First term: \(125x^3\)
• Second term: \(-75x^2\)
• Third term: \(15x\)
• Fourth term: \(-1\)

Algebraic expressions allow for general formulations and solutions to various mathematical problems, giving us tools to reformat and analyze equations.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations of objects. It's fundamental to solving problems involving counting and arrangements, like calculating binomial coefficients used in polynomial expansions.

In the Binomial Theorem, combinatorics is applied to determine the coefficients \(\binom{n}{k}\):

• \((3 \text{ choose } 0) = 1\)
• \((3 \text{ choose } 1) = 3\)
• \((3 \text{ choose } 2) = 3\)
• \((3 \text{ choose } 3) = 1\)

These coefficients indicate the number of ways to select terms while respecting order and repetitions. It's like choosing toppings for a sundae, but with math!

By utilizing combinatorial principles, we can break down complex expressions into manageable parts and ensure our solutions are precise. Combinatorics not only supports algebraic operations but also enhances our ability to solve diverse mathematical challenges.