Expand (x-y)^4 Using The Fourth Terms In The Binomial Expansion

Expand (x-y)^4 Using The Fourth Terms In The Binomial Expansion

Expand (x-y)^4 Using The Fourth Terms In The Binomial Expansion is a common mathematical problem that involves applying the binomial theorem to simplify and expand algebraic expressions. This process not only enhances your understanding of polynomial expansion but also sharpens your skills in algebraic manipulation. In this article, we will explore the binomial expansion method, identify the fourth terms in the expansion, and demonstrate how to expand (x - y)^4 step-by-step.

Understanding the Binomial Theorem

What Is the Binomial Theorem?

The binomial theorem provides a formula for expanding powers of binomials, which are algebraic expressions with two terms, such as (a + b) or (x - y). The general form of the binomial theorem for any positive integer n is:

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

where:

• \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).


• \(a\) and \(b\) are the terms in the binomial.

Application to (x - y)^4

In our case, the binomial is (x - y), and n=4. Using the binomial theorem, the expansion becomes:

\[
(x - y)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} (-y)^k
\]

Notice that the negative sign applies to y, which affects the signs of the terms in the expansion.

Calculating the Fourth Terms in the Binomial Expansion

Understanding the Terms of the Expansion

The terms in the binomial expansion are arranged based on the value of k, starting from 0 up to n. For (x - y)^4, the terms are:

• Term 1: \(k=0\)
• Term 2: \(k=1\)
• Term 3: \(k=2\)
• Term 4: \(k=3\)
• Term 5: \(k=4\)

Since the question emphasizes the "Fourth Terms," it is essential to clarify whether we mean the fourth term in the sequence or specifically the term corresponding to \(k=3\). Typically, the "fourth term" refers to the term where \(k=3\).

Identifying the Fourth Term

The fourth term in the expansion corresponds to \(k=3\), and its general form is:

\[
T_{k+1} = \binom{n}{k} a^{n-k} b^{k}
\]

Applying to our case:

\[
T_{4} = \binom{4}{3} x^{4-3} (-y)^3
\]

Calculating each component:

• \(\binom{4}{3} = 4\)


• \(x^{1} = x\)


• \((-y)^3 = - y^3\)

Putting it all together:

\[
T_{4} = 4 \times x \times (- y^3) = -4 x y^3
\]

This is the fourth term in the binomial expansion of (x - y)^4.

Full Expansion of (x - y)^4

While the focus is on the fourth term, understanding the entire expansion provides better context. Let's calculate all the terms for (x - y)^4.

Step-by-Step Expansion

Using the binomial theorem:

\[
(x - y)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} (- y)^k
\]

Calculating each term:

• k=0:

\[
\binom{4}{0} x^{4} (- y)^0 = 1 \times x^4 \times 1 = x^4
\]

• k=1:

\[
\binom{4}{1} x^{3} (- y)^1 = 4 \times x^3 \times (- y) = -4 x^3 y
\]

• k=2:

\[
\binom{4}{2} x^{2} (- y)^2 = 6 \times x^2 \times y^2 = 6 x^2 y^2
\]

• k=3:

\[
\binom{4}{3} x^{1} (- y)^3 = 4 \times x \times (- y)^3 = -4 x y^3
\]

• k=4:

\[
\binom{4}{4} x^{0} (- y)^4 = 1 \times 1 \times y^4 = y^4
\]

Complete expanded form:

\[
x^4 - 4 x^3 y + 6 x^2 y^2 - 4 x y^3 + y^4
\]

Interpreting the Fourth Term in the Expansion

Significance of the Fourth Term

The fourth term, \(-4 x y^3\), plays a crucial role in understanding the behavior of the polynomial, especially in applications involving polynomial identities, algebraic factoring, and binomial coefficients.

Properties of the Fourth Term

• It has a negative sign, indicating the alternating nature of the binomial expansion.
• The term involves the variables \(x\) and \(y\) with powers 1 and 3 respectively.
• The coefficient 4 is derived from the binomial coefficient \(\binom{4}{3}\).

Applications of Binomial Expansion in Mathematics

Algebraic Simplification

Expanding binomials is essential for simplifying complex algebraic expressions, especially in polynomial multiplication.

Calculating Probabilities

Binomial coefficients are used extensively in probability theory, especially in binomial probability calculations.

Combinatorial Analysis

Understanding the binomial expansion helps in counting problems and combinatorial analysis.

Practice Problems

To solidify understanding, here are some practice problems related to binomial expansion:

• Expand (x + y)^4 and identify the third term.
• Find the coefficient of the \(x^2 y^2\) term in the expansion of (x - y)^4.
• Verify the expansion of (x - y)^4 by direct multiplication.
• Using the binomial theorem, expand (2x - 3y)^4.
• Determine the fourth term in the expansion of (a + b)^5.

Conclusion

Understanding how to expand (x - y)^4 using the fourth terms in the binomial expansion is foundational in algebra. The process involves applying the binomial theorem, calculating binomial coefficients, and carefully expanding each term with the correct signs. Recognizing the significance of individual terms, especially the fourth term, enhances your ability to manipulate algebraic expressions and solve complex problems efficiently. Whether for academic purposes or practical applications, mastering binomial expansion is an essential skill in mathematics.

Summary of Key Points

• The binomial theorem provides a formula for expanding powers of binomials.
• The fourth term corresponds to \(k=3\) in the expansion of (x - y)^4.
• The fourth term is \(-4 x y^3\).
• The entire expansion of (x - y)^4 is \(x^4 - 4 x^3 y + 6 x^2 y^2 - 4 x y^3 + y^4\).
• Understanding individual terms helps in algebraic simplification and problem-solving.

By mastering the technique of binomial expansion and focusing on specific terms, you can tackle a wide range of algebraic and combinatorial problems with confidence.