Problem 77 - Algebra 2

Question

Expand each binomial. $$ (3 x-5)^{6} $$

Step-by-Step Solution

Answer

The expansion of \( (3x-5)^6 \) simplifies to \( 729x^6 - 8748x^5 + 50625x^4 - 150000x^3 + 225000x^2 -167500x + 15625 \).

Step 1: Identify the components of the binomial and the power

The task is to expand the binomial \( (3x-5)^6 \). Here, \(x = 3x\), \(y = -5\), and \(n = 6\).

Step 2: Apply the Binomial Theorem

Apply the formula of Binomial Theorem for every \(k\) from 0 to 6. This will give the following expressions: \n For \(k = 0\), the term is \(6C0 * (3x)^6 * (-5)^0\),For \(k = 1\), the term is \(6C1 * (3x)^5 * (-5)^1\),For \(k = 2\), the term is \(6C2 * (3x)^4 * (-5)^2\),For \(k = 3\), the term is \(6C3 * (3x)^3 * (-5)^3\),For \(k = 4\), the term is \(6C4 * (3x)^2 * (-5)^4\),For \(k = 5\), the term is \(6C5 * (3x)^1 * (-5)^5\),For \(k = 6\), the term is \(6C6 * (3x)^0 * (-5)^6\).\n Add all these terms to expand the given binomial.

Step 3: Simplify each term

Compute each term individually. This involves calculating binomial coefficients with 'n choose k', simplifying powers, and performing multiplications and additions. The resulting expansion should be simplified as much as possible.

Step 4: Write down the simplified expression

Combine all of the terms together to form the expanded binomial. Make sure to simplify the coefficients wherever possible and order the terms in descending powers of \(x\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra used to expand expressions of the form \((a + b)^n\). It allows us to represent these expressions as a series of terms in a polynomial sum. The theorem states that:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

Here, \(\binom{n}{k}\) represents the binomial coefficients, defining how many ways we can choose \(k\) elements from a set of \(n\) elements.
In the binomial expansion, each term is a product of a binomial coefficient, a power of the first term, and a power of the second term. This makes it highly systematic and predictable when expanding binomials.

Polynomial Expansion

Polynomial expansion is the process of expressing a binomial raised to a power as a sum of individual monomial terms. When a binomial like \((3x - 5)^6\) is expanded, each resulting term comes from using the binomial theorem.
The steps in polynomial expansion are fairly straightforward:

• Identify the binomial and its power.
• Determine each term by using the formula from the binomial theorem.
• Combine all terms to form a cohesive and simplified polynomial.

This systematic approach allows any binomial to be easily expanded into its polynomial form, making it a vital algebraic tool.

Binomial Coefficients

Binomial coefficients are pivotal in calculating the terms of a binomial expansion. They are represented as \(\binom{n}{k}\), known as "n choose k". These coefficients can be calculated using the formula:

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

where \(!\) denotes factorial, meaning the product of all positive integers up to that number.
For instance, when expanding \((3x - 5)^6\), the coefficient for any term is determined by how many ways \(k\) successes can occur in \(n\) trials of binomials. This combinatorial nature of binomial coefficients makes them a central part of understanding polynomial expansions.

Algebraic Expressions

Algebraic expressions consist of variables and constants, combined using operations such as addition, subtraction, multiplication, and exponentiation. A binomial, such as \((3x - 5)\), is an algebraic expression containing exactly two terms.
In the context of binomial expansions, we manipulate these algebraic expressions using rules and theorems to simplify or expand them. Understanding algebraic expressions is crucial, as it forms the basis of working with higher-level mathematics.

• Variables represent unknown quantities that can vary.
• Constants are fixed values.
• Operations regulate how these pieces come together.

The elegance of algebraic expressions lies in their versatility, allowing a wide range of mathematical explorations and solutions.