Problem 28 - College Algebra

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x-3 y)^{5} $$

Step-by-Step Solution

Answer

The expanded form of \((x-3 y)^{5}\) is \(x^5-15x^4y+225x^3y^2-2025x^2y^3+12150xy^4-24300y^5\).

Step 1: Identify the Binomial Form

The given expression \((x-3 y)^{5}\) is in the binomial form \((a-b)^{n}\) where \(a=x\), \(b=3y\) and \(n=5\).

Step 2: Apply the Binomial Theorem

Next, apply the Binomial theorem to expand the given binomial. The Binomial Theorem for \((a - b)^n\) is \(\sum _{k=0} ^{n} (-1)^k \binom{n}{k} a^{n-k}b^{k}\)

Step 3: Calculate Each Term

When the Binomial theorem is applied to the expression, it expands to \( \binom{5}{0}x^{5}(-3y)^{0}-\binom{5}{1}x^{4}(-3y)^{1}+\binom{5}{2}x^{3}(-3y)^{2}-\binom{5}{3}x^{2}(-3y)^{3}+\binom{5}{4}x^{1}(-3y)^{4}-\binom{5}{5}x^{0}(-3y)^{5}\)

Step 4: Simplify the Binomial Coefficients and Negative Signs

Now simplify each term by calculating the binomial coefficients and simplifying the negative signs. This simplifies to \(x^5-15x^4y+225x^3y^2-2025x^2y^3+12150xy^4-24300y^5\)

Step 5: Final Solution

After computing each term, the final solution is \(x^5-15x^4y+225x^3y^2-2025x^2y^3+12150xy^4-24300y^5\)

Key Concepts

Simplify Binomial Expressions

Understanding how to simplify binomial expressions is essential in algebra. A binomial is a polynomial with exactly two terms. When raising a binomial to a power, such as \( (x-3y)^5 \), we should simplify the expression to make it easier to work with. To do this, we use the Binomial Theorem, which provides a formula for expanding binomials raised to any power.

The process includes identifying the two terms of the binomial and then using the formula to expand it systematically. After applying the theorem, we often need to combine like terms and carry out any necessary multiplications or simplifications. For example, after expanding, negative signs and binomial coefficients must be addressed, ensuring that our expression is in its simplest form, as seen in the final solution: \( x^5-15x^4y+225x^3y^2-2025x^2y^3+12150xy^4-24300y^5 \).

Binomial Coefficients

Binomial coefficients are the numbers that appear as coefficients in the Binomial Theorem expansion. They can be denoted by \(\binom{n}{k}\), representing the number of ways to choose \(k\) elements from a set of \(n\) without regard to order. Also known as 'combination' or 'choose' numbers, they are found in Pascal's Triangle or calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) is the factorial of \(n\).

When expanding \( (x-3y)^5 \), we compute binomial coefficients like \(\binom{5}{0}\), \(\binom{5}{1}\), and so on, each of which multiplies a corresponding term in the expansion. Their correct calculation is critical to achieving the accurate expanded form of the binomial expression.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers using arithmetic operations—addition, subtraction, multiplication, division, and exponentiation. The letters are variables representing certain values. In the case of the binomial \( (x-3y)^5 \), the terms \(x\) and \(3y\) represent algebraic expressions which can take on specific values.

The power of algebra lies in the ability to work with these variable expressions and manipulate them according to algebraic rules and theorems, such as the Binomial Theorem. This theorem is a powerful tool that turns a complex operation, raising a binomial to a power, into a series of simpler operations, combining the power of both algebra and combinatorics.

College Algebra

College Algebra is an advanced course that includes the study of functions, complex numbers, and the depths of polynomial equations. The Binomial Theorem is an important topic within this course, as it extends the range of algebraic expressions and equations that students can solve. Mastery of this theorem empowers students to handle polynomial expressions more efficiently and understand the structure of algebraic expansions.

The exercise of expanding \( (x-3y)^5 \) using the Binomial Theorem is a classic example of College Algebra's application in simplifying complex algebraic expressions. The skills practiced here are foundational for higher-level mathematics and various applications within scientific and engineering fields.