Problem 100 - Precalculus with Limits

Question

How do the expansions of \( \left(x + y\right)^n \) and \( \left(x - y\right)^n \) differ? Support your explanation withan example.

Step-by-Step Solution

Answer

The expansions of the two expressions \( \left(x + y\right)^n \) and \( \left(x - y\right)^n \) predominantly differ in the signs of the terms with odd powers of y. In \( \left(x - y\right)^n \), there is a minus sign for the terms of the expansion that have an odd exponent of y, as opposed to \( \left(x + y\right)^n \) where all terms are positive.

Step 1: Expanding \( \left(x + y\right)^n \)

To expand this expression, let's use the binomial theorem, which states that \( (x + y)^n = \Sigma_{k=0}^n C(n, k) * x^{n-k} * y^{k} \), where \( C(n, k) \) are the binomial coefficients. If n=3, for example, the expansion would be: \( x^3 + 3x^2y + 3xy^2 + y^3 \).

Step 2: Expanding \( \left(x - y\right)^n \)

Similarly, we use the binomial theorem again. However here the 'y' term is subtracted altogether which gives a minus sign for the terms of the expansion that have an odd exponent of y. For the example given above with n=3, the expansion would be: \( x^3 - 3x^2y + 3xy^2 - y^3 \).

Step 3: Comparing the Expansions

Upon comparing the expansions of \( \left(x + y\right)^n \) and \( \left(x - y\right)^n \), it can be observed that the sign in every odd power of y term will flip in the expansion of \( \left(x - y\right)^n \) as compared to \( \left(x + y\right)^n \). This happens because of the alternate positive and negative signs in the expansion of \( \left(x - y\right)^n \).

Key Concepts

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial raised to a power in a more expanded form. It allows us to write expressions like \((x + y)^n\) or \((x - y)^n\) in a format where terms are separated and any exponents are applied. This expansion can simplify working with polynomials, particularly when you need to add, subtract, or evaluate them.

Think of it like unpacking a box. The initial expression \((x+y)^n\) is compact, but polynomial expansion lets us "unpack" it into terms that show every separate piece of information. This is crucial for many algebra problems and helps in manipulating and understanding the polynomial better.

• Using expansion, especially for higher powers, makes it easier to compute or simplify equations.
• It provides a clearer view of the structure of the polynomial, its individual terms, and coefficients.

Understanding polynomial expansion is foundational in algebra and helps when dealing with more complex mathematical operations.

Binomial Coefficients

Binomial coefficients are key figures in the binomial expansion process. They represent the number of ways to choose a certain number of elements from a given set, mathematically shown as \(C(n, k)\). In the context of polynomial expansion, they tell us how to weight each term.

For example, in the expansion of \((x + y)^3= x^3 + 3x^2y + 3xy^2 + y^3\), the coefficients 1, 3, 3, 1 come from the row of Pascal's Triangle for \(n=3\). Each coefficient corresponds to the different ways terms can combine in the expression.

• These coefficients determine the distribution of terms in the expansion.
• Binomial coefficients are symmetric, and the triangle is an easy way to see the pattern.

This concept is a crucial part of algebra and combinatorics, unlocking the structure within expanded polynomials.

Power of a Binomial

When we talk about the power of a binomial, we refer to expressions like \((x + y)^n\) where the binomial \((x + y)\) is raised to a certain power \(n\). The power of a binomial is an essential part of understanding its expansion and behavior.

The power dictates how many terms will be in the expansion and how complex those terms will be. In each step of expansion, the exponents of \(x\) and \(y\) change, and the binomial coefficients distribute them according to the binomial theorem. Thus, the power of a binomial is not just about multiplying terms, but about uncovering a pattern that repeats itself predictably.

• The higher the power, the more terms there will be in the expansion.
• The alternating pattern of signs in \((x - y)^n\) is due to the subtraction in the binomial base, affecting every term with an odd exponent.

Understanding this concept is significant for solving polynomial equations and analyzing mathematical problems with binomials.