Problem 37 - Algebra for College Students

Question

Expand each binomial and simplify.\((x-y)^{8}\)

Step-by-Step Solution

Answer

\((x-y)^8 = x^8 - 8x^7y + 28x^6y^2 - 56x^5y^3 + 70x^4y^4 - 56x^3y^5 + 28x^2y^6 - 8xy^7 + y^8\)

Step 1: Identify the Binomial Expression

The given binomial expression is \((x - y)^8\). This expression needs to be expanded and simplified.

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that for any positive integer \(n\), \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). Here, \(a = x\), \(b = -y\), and \(n = 8\). Substitute these values into the theorem.

Step 3: Calculate Binomial Coefficients

Use the formula for binomial coefficients \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) to calculate for all \(k\) from 0 to 8. The coefficients for \(\binom{8}{k}\) are 1, 8, 28, 56, 70, 56, 28, 8, 1.

Step 4: Write the Expanded Expression

Use the coefficients calculated to write the terms of the expansion: \(x^8 - 8x^7y + 28x^6y^2 - 56x^5y^3 + 70x^4y^4 - 56x^3y^5 + 28x^2y^6 - 8xy^7 + y^8\).

Step 5: Simplify the Expansion

Since the expression is already expanded into its simplest form with each term having a distinct combination of powers of \(x\) and \(y\), no further simplification is needed.

Key Concepts

Binomial Theorem

The Binomial Theorem is a formula describing the algebraic expansion of powers of a binomial. A binomial is an algebraic expression containing two terms connected by a plus or a minus sign. For example, (x - y) is a binomial. The theorem tells us how to expand expressions like ext{(a + b)}^n into a sum involving terms of the form {a} \cdot {b} However, when involving powers, it includes coefficients.

This theorem is written as: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \).

Using this, we can take any two-variable power, no matter how big, and break it down into manageable terms by calculating the binomial coefficients and using the power sequence of two given variables. This makes polynomial expressions more attainable and is very useful in algebra and calculus for proving various concepts or simplifying expressions.

binomial coefficients

Binomial coefficients are the heart of the Binomial Theorem. They provide the necessary weights to each term in the expansion, determining how many times each term appears in the expanded polynomial. Calculating these coefficients involves a simple combination formula given by:
\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
Where:

• n is the power to which the binomial is raised.
• k is the specific term in the sequence.

In our example of (x-y)^8\ : the coefficients are 1, 8, 28, 56, 70, 56, 28, 8, and 1, calculated for k values from 0 to 8. You can find these coefficients in the "Pascal's Triangle," where each row corresponds to the coefficients of a binomial expansion for sequential powers.

polynomial simplification

Polynomial simplification involves combining like terms and expressing the polynomial in its simplest form. When working with binomial expansions, simplification ensures that the expansion is practical for solving equations or understanding its behavior.
In the expanded version of (x-y)^8\ :
\(x^8 - 8x^7y + 28x^6y^2 - 56x^5y^3 + 70x^4y^4 - 56x^3y^5 + 28x^2y^6 - 8xy^7 + y^8\),
no further simplification is possible as all terms have distinct powers of x and y. The polynomial is already organized in a standard descending order of the powers of one variable. It's crucial to recognize when an expression is in its simplest form to avoid unnecessary steps and errors.