Problem 3 - Algebra 2

Question

Expand each power. $$ (x-3 y)^{4} $$

Step-by-Step Solution

Answer

The expansion is \(x^4 - 12x^3y + 54x^2y^2 - 108xy^3 + 81y^4\).

Step 1: Identify the Formula for Binomial Expansion

The expression \((x - 3y)^4\) can be expanded using the Binomial Theorem. This theorem states that \((a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\). In this expression, \(a = x\), \(b = -3y\), and \(n = 4\).

Step 2: Determine Each Term of the Expansion

To find each term of the expansion, we need to calculate \({4 \choose k} x^{4-k} (-3y)^k\) for \(k\) ranging from 0 to 4. This involves calculating the binomial coefficients and powers step by step.

Step 3: Calculate Each Term

- For \(k = 0\): \[{4 \choose 0} x^4 (-3y)^0 = 1 \cdot x^4 = x^4\] - For \(k = 1\): \[{4 \choose 1} x^3 (-3y)^1 = 4 \cdot x^3 \cdot (-3y) = -12x^3y\]- For \(k = 2\): \[{4 \choose 2} x^2 (-3y)^2 = 6 \cdot x^2 \cdot 9y^2 = 54x^2y^2\]- For \(k = 3\): \[{4 \choose 3} x^1 (-3y)^3 = 4 \cdot x \cdot (-27y^3) = -108xy^3\]- For \(k = 4\): \[{4 \choose 4} x^0 (-3y)^4 = 1 \cdot 81y^4 = 81y^4\]

Step 4: Combine the Terms

Now, combine the terms calculated from each step into a single expression. The expanded form of \((x-3y)^4\) is:\[ x^4 - 12x^3y + 54x^2y^2 - 108xy^3 + 81y^4 \]

Key Concepts

Binomial Expansion

The binomial expansion is a way to express the power of a binomial as a sum. Using the Binomial Theorem, a binomial expression like \((x - 3y)^4\) can be expanded into a series of terms. Each term represents a part of the original expression raised to various powers, with coefficients that give each term its proper weight.
To perform a binomial expansion, we use the formula \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). This allows us to break down the binomial into manageable parts.

• a and b are the terms in the binomial,
• n is the exponent indicating the power of the binomial.

This method makes it easier to understand how each component of the binomial contributes to the final expanded form. By expanding, we not only distribute the power across both terms but also illustrate their interaction.

Binomial Coefficients

Binomial coefficients are crucial in the process of binomial expansion. These coefficients are represented by the notation \(\binom{n}{k}\), which reads as "n choose k" and stands for the number of ways to choose \(k\) elements from \(n\) elements without regard to order. It’s also known as a combination.

For the expression \((x - 3y)^4\), the coefficients are:\

• \(\binom{4}{0} = 1\)
• \(\binom{4}{1} = 4\)
• \(\binom{4}{2} = 6\)
• \(\binom{4}{3} = 4\)
• \(\binom{4}{4} = 1\)

These coefficients directly multiply with the terms generated by raising a and b to their respective powers, producing terms like \(x^4\) or \(-12x^3y\). Understanding these coefficients helps predict the behavior of the expanded form.

Polynomial Expansion

Polynomial expansion involves writing a polynomial in its explicit form. In the context of binomial expansion, it means expanding a binomial expression such as \((x - 3y)^4\) into a sum of terms.

The expanded polynomial is the sum of terms like \(x^4 - 12x^3y + 54x^2y^2 - 108xy^3 + 81y^4\). Each term is calculated using:

• Power of \(a\) and \(b\)
• Multiplication with binomial coefficients
• Proper sign assignment, especially when terms are negative.

Polynomial expansion finds widespread use in algebra and calculus, as it simplifies complex expressions and helps in polynomial approximations and manipulations.

Algebra 2

In Algebra 2, understanding binomial expansions is crucial as it forms the basis for other advanced topics. It connects with concepts such as permutations, combinations, and probability.

In a typical Algebra 2 course, students would encounter the Binomial Theorem and its application to polynomials, which provides foundational skills for studying advanced algebra topics and calculus.

• Helps in solving complex polynomial equations
• Enhances comprehension of sequences and series
• Builds a foundation for learning calculus.

Thus, mastering the binomial expansion in Algebra 2 is not only about solving individual problems but also preparing for more advanced mathematical concepts.