Expanding (2x - Y)^3 A Step-by-Step Guide

In mathematics, expanding polynomial expressions is a fundamental skill. One common type of expression involves raising a binomial (an expression with two terms) to a power. In this article, we will focus on expanding the expression $(2x - y)^3$. This involves using the binomial theorem or direct multiplication to find the equivalent polynomial. We will explore the steps involved, common pitfalls to avoid, and verify our results. Mastering this type of algebraic manipulation is crucial for success in higher-level mathematics, including calculus and linear algebra. Let’s dive into the process of expanding $(2x - y)^3$ and understanding the underlying principles.

At the heart of our discussion is the expansion of the binomial expression $(2x - y)^3$. This means we need to multiply the expression $(2x - y)$ by itself three times. There are two primary methods to achieve this: the binomial theorem and direct multiplication. Both methods will yield the same result, but understanding each approach can provide a more comprehensive grasp of polynomial expansion. Before we delve into the step-by-step solutions, let's briefly discuss these two methods.

The binomial theorem provides a formula for expanding expressions of the form $(a + b)^n$, where n is a non-negative integer. This theorem utilizes binomial coefficients, which can be found using Pascal's Triangle or the combination formula. The binomial theorem is particularly useful for expanding expressions with higher powers, as it provides a systematic approach to the expansion. However, for smaller powers like 3, direct multiplication can often be just as efficient.

Direct multiplication involves multiplying the binomial $(2x - y)$ by itself multiple times. First, we multiply $(2x - y)$ by $(2x - y)$ to get $(2x - y)^2$. Then, we multiply the result by $(2x - y)$ again to obtain $(2x - y)^3$. This method is straightforward and relies on the distributive property of multiplication. While it may be more tedious for higher powers, it offers a concrete understanding of how each term in the expansion is derived. Choosing the right method depends on the complexity of the expression and personal preference. In the following sections, we will demonstrate both methods to ensure a thorough understanding of the expansion process.

The binomial theorem provides a systematic way to expand expressions of the form $(a + b)^n$. In our case, we have $(2x - y)^3$, so $a = 2x$, $b = -y$, and $n = 3$. The binomial theorem formula is:

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

where $\binom{n}{k}$ represents the binomial coefficient, also known as "n choose k", which can be calculated as:

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

For our specific problem, $(2x - y)^3$, we need to find the coefficients for $k = 0, 1, 2,$ and $3$.

Let's calculate these binomial coefficients:

• For $k = 0$: $\binom{3}{0} = \frac{3!}{0!(3-0)!} = \frac{3!}{0!3!} = 1$
• For $k = 1$: $\binom{3}{1} = \frac{3!}{1!(3-1)!} = \frac{3!}{1!2!} = \frac{3 \times 2 \times 1}{(1)(2 \times 1)} = 3$
• For $k = 2$: $\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3!}{2!1!} = \frac{3 \times 2 \times 1}{(2 \times 1)(1)} = 3$
• For $k = 3$: $\binom{3}{3} = \frac{3!}{3!(3-3)!} = \frac{3!}{3!0!} = 1$

Now we can apply the binomial theorem formula:

$(2x - y)^3 = \binom{3}{0}(2x)^3(-y)^0 + \binom{3}{1}(2x)^2(-y)^1 + \binom{3}{2}(2x)^1(-y)^2 + \binom{3}{3}(2x)^0(-y)^3$

Substitute the binomial coefficients we calculated:

$(2x - y)^3 = 1(2x)^3(-y)^0 + 3(2x)^2(-y)^1 + 3(2x)^1(-y)^2 + 1(2x)^0(-y)^3$

Now, simplify each term:

• $1(2x)^3(-y)^0 = 1(8x^3)(1) = 8x^3$
• $3(2x)^2(-y)^1 = 3(4x^2)(-y) = -12x^2y$
• $3(2x)^1(-y)^2 = 3(2x)(y^2) = 6xy^2$
• $1(2x)^0(-y)^3 = 1(1)(-y^3) = -y^3$

Combine the terms:

$(2x - y)^3 = 8x^3 - 12x^2y + 6xy^2 - y^3$

Thus, using the binomial theorem, we have expanded $(2x - y)^3$ to $8x^3 - 12x^2y + 6xy^2 - y^3$. This result provides a clear understanding of how the binomial theorem works and how it can be applied to expand binomial expressions efficiently.

Direct multiplication involves multiplying the expression $(2x - y)$ by itself three times. This method relies on the distributive property of multiplication and can be a straightforward way to expand binomial expressions. Let's break down the steps:

First, we multiply $(2x - y)$ by $(2x - y)$ to obtain $(2x - y)^2$:

$(2x - y)(2x - y) = (2x)(2x) + (2x)(-y) + (-y)(2x) + (-y)(-y)$

$= 4x^2 - 2xy - 2xy + y^2$

$= 4x^2 - 4xy + y^2$

So, $(2x - y)^2 = 4x^2 - 4xy + y^2$.

Next, we multiply the result $(4x^2 - 4xy + y^2)$ by $(2x - y)$ to get $(2x - y)^3$:

$(4x^2 - 4xy + y^2)(2x - y) = 4x^2(2x - y) - 4xy(2x - y) + y^2(2x - y)$

Now, distribute each term:

$= 4x^2(2x) + 4x^2(-y) - 4xy(2x) - 4xy(-y) + y^2(2x) + y^2(-y)$

$= 8x^3 - 4x^2y - 8x^2y + 4xy^2 + 2xy^2 - y^3$

Combine like terms:

$= 8x^3 - 4x^2y - 8x^2y + 4xy^2 + 2xy^2 - y^3$

$= 8x^3 - 12x^2y + 6xy^2 - y^3$

Thus, by direct multiplication, we have expanded $(2x - y)^3$ to $8x^3 - 12x^2y + 6xy^2 - y^3$. This method provides a clear, step-by-step approach to expanding the expression, demonstrating how each term is derived through multiplication and combination.

Both the binomial theorem and direct multiplication methods have successfully expanded the expression $(2x - y)^3$ to $8x^3 - 12x^2y + 6xy^2 - y^3$. Understanding both methods is valuable, as each offers unique insights into polynomial expansion.

The binomial theorem provides a systematic and efficient way to expand binomial expressions, especially for higher powers. It relies on the binomial coefficients, which can be calculated using combinations or Pascal's Triangle. This method is advantageous when dealing with large exponents, as it avoids the tedious process of repeated multiplication. However, it requires familiarity with the binomial theorem formula and the calculation of binomial coefficients.

Direct multiplication, on the other hand, involves multiplying the binomial by itself repeatedly. This method is more intuitive and relies on the distributive property of multiplication. It is particularly useful for smaller powers, such as the cube in our example. Direct multiplication provides a clear, step-by-step understanding of how each term in the expansion is derived. However, it can become cumbersome and error-prone for higher powers, as the number of terms to multiply and combine increases significantly.

In summary, the choice between the binomial theorem and direct multiplication depends on the specific problem and personal preference. The binomial theorem is more efficient for higher powers, while direct multiplication offers a more concrete understanding of the expansion process, especially for smaller powers. Both methods are essential tools in algebra, and mastering them will enhance your ability to manipulate polynomial expressions effectively.

Having expanded $(2x - y)^3$ using both the binomial theorem and direct multiplication, we arrived at the result $8x^3 - 12x^2y + 6xy^2 - y^3$. Now, let’s compare this result with the provided options to identify the correct answer.

The options given are:

A. $6x^3 - 18x^2y - 2xy^2 - 8y^3$

B. $-6x^3 + 6x^2y - 5xy^2 + y^3$

C. $8x^3 - 12x^2y + 6xy^2 - y^3$

D. $-4x^3 + 16x^2y + 4xy^2 + 8y^3$

By comparing our expanded expression $8x^3 - 12x^2y + 6xy^2 - y^3$ with the given options, it is clear that option C matches our result exactly. Therefore, option C is the correct answer.

This step highlights the importance of verifying the expanded expression with the provided options to ensure accuracy. It also reinforces the value of having a clear, step-by-step approach to expansion, whether using the binomial theorem or direct multiplication, to minimize errors.

When expanding expressions like $(2x - y)^3$, it's easy to make mistakes if you're not careful. Identifying and understanding these common errors can significantly improve accuracy. Here are some pitfalls to watch out for:

1. Incorrectly Applying the Distributive Property: A common mistake is not distributing terms correctly during multiplication. For example, when multiplying $(4x^2 - 4xy + y^2)$ by $(2x - y)$, ensure each term in the first expression is multiplied by each term in the second expression. This requires careful attention to detail and can be avoided by systematically distributing each term.

2. Sign Errors: Pay close attention to the signs, especially when dealing with negative terms like $-y$. For example, when squaring $-y$, it becomes $y^2$, but when cubing it, it becomes $-y^3$. Incorrectly handling signs can lead to significant errors in the final result.

3. Forgetting to Combine Like Terms: After expanding, it's crucial to combine like terms to simplify the expression. For instance, terms like $-4x^2y$ and $-8x^2y$ should be combined to $-12x^2y$. Overlooking this step can result in an incomplete or incorrect answer.

4. Miscalculating Binomial Coefficients: When using the binomial theorem, errors in calculating binomial coefficients can occur. Ensure the correct formula is applied, and the factorials are computed accurately. Double-checking these calculations can prevent mistakes.

5. Incorrectly Applying Exponents: When raising terms to powers, ensure both the coefficient and the variable are raised to the power. For example, $(2x)^3$ is $8x^3$, not $2x^3$. Paying attention to the exponent's effect on all parts of the term is essential.

By being aware of these common mistakes and taking the time to double-check each step, you can enhance your accuracy and confidence in expanding polynomial expressions.

In conclusion, expanding the expression $(2x - y)^3$ involves applying either the binomial theorem or direct multiplication. Both methods yield the same result, which is $8x^3 - 12x^2y + 6xy^2 - y^3$. The binomial theorem provides a systematic approach using binomial coefficients, while direct multiplication involves multiplying the binomial by itself multiple times, relying on the distributive property.

Understanding both methods is crucial for mastering polynomial expansion. The binomial theorem is particularly useful for higher powers, while direct multiplication offers a clear, step-by-step understanding for smaller powers. By comparing our expanded expression with the given options, we identified option C as the correct answer.

Avoiding common mistakes, such as incorrectly applying the distributive property, sign errors, forgetting to combine like terms, miscalculating binomial coefficients, and incorrectly applying exponents, is essential for accuracy. By being mindful of these pitfalls and double-checking each step, you can improve your ability to expand polynomial expressions confidently.

Mastering these algebraic techniques is fundamental for success in various areas of mathematics, including calculus, linear algebra, and beyond. Practice and familiarity with these concepts will enable you to tackle more complex problems with ease and precision.