Problem 34 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(x^2 + y^2\right)^6 \)

Step-by-Step Solution

Answer

The expansion and simplification of the given expression using the binomial theorem is: \[ x^{12} + 6x^{10}y^2 + 15x^8y^4 + 20x^6y^6 + 15x^4y^8 + 6x^2y^{10} + y^{12} \]

Step 1: Recognize the binomial theorem

The binomial theorem states that for any real numbers x and y, \((x + y)^n\) can be expanded into a sum involving terms of the form \(nCk * x^{n-k} * y^{k}\). In this case, x is \(x^2\), y is \(y^2\), and n is 6.

Step 2: Expand the binomial using the theorem

We will apply the binomial theorem and expand the expression. Since n is 6, there will be seven terms in the expansion after adding 1 to n. The coefficients for the terms are obtained from the 6th row of Pascal's Triangle or from the binomial coefficient formula \(6Ck\). The nth term in the expansion is given by the formula \(6Ck * (x^2)^{6-k} * (y^2)^{k}\), where k ranges from 0 to 6.

Step 3: Simplify the expanded binomial

Since each term of our binomial raises \(x^2\) and \(y^2\) to certain powers, we can simplify these to get terms involving \(x\) and \(y\) to even powers only.

Key Concepts

Pascal's Triangle

Pascal's Triangle is a fascinating and useful tool in mathematics, especially when dealing with binomial expansions. This triangle starts with a top row having just a single '1'. Each following row is constructed by adding the number above and to the left with the number above and to the right, assuming non-existent positions as '0'.
For example, the third row is formed as follows:
\[1, \ 2, \ 1\] which corresponds to the coefficients of \((x + y)^2\).
To find the coefficients for the expansion \((x^2 + y^2)^6\), look at the 6th row:
\[1, \ 6, \ 15, \ 20, \ 15, \ 6, \ 1\]
These numbers are the coefficients you’ll use in your expansion. Each number represents a coefficient for a term in the expansion. This method is quick and visual, helping you easily find coefficients without calculations.

Binomial Expansion

The Binomial Expansion is a crucial concept for expanding expressions raised to a power, described by the Binomial Theorem.
This theorem states that any binomial expression \((x + y)^n\) can be expanded into a series of terms in the format:

• \(nCk\)
• \(x^{n-k}\)
• \(y^k\)

The task is to apply this for specific values. In our case, \((x^2 + y^2)^6\) is expanded by treating \(x^2\) as \(x\) and \(y^2\) as \(y\), with \(n = 6\).
Thus, each term in the expansion is formed using the formula:
\[6Ck \times (x^2)^{6-k} \times (y^2)^k\]
The binomial expansion results in a sum of terms, each involving a power of \(x^2\) and \(y^2\), easily convertible to even powers of \(x\) and \(y\). This systematic way ensures no term is missed, keeping the expansion correct and complete.

Coefficients

Coefficients are numerical factors in the terms of a polynomial expansion. In the context of the Binomial Theorem, they are obtained using binomial coefficients, calculated via Pascal's Triangle or through the formula \(nCk\), which represents a combination or a specific number of ways to choose \(k\) elements from \(n\) options.
For \((x^2 + y^2)^6\), coefficients are:

• \(6C0 = 1\)
• \(6C1 = 6\)
• \(6C2 = 15\)
• \(6C3 = 20\)
• \(6C4 = 15\)
• \(6C5 = 6\)
• \(6C6 = 1\)

These coefficients multiply the terms \((x^2)^{6-k}\) and \((y^2)^k\) to form each part of the expanded expression.
Understanding how these coefficients work is essential for correctly performing binomial expansions, ensuring each term’s place and value is accurate in the full expression.