Question

Verify each expansion. Obtain the binomial coefficients by formula or from Pascal's triangle as directed by your instructor. $$\left(1 / x+1 / y^{2}\right)^{3}=1 / x^{3}+3 / x^{2} y^{2}+3 / x y^{4}+1 / y^{6}$$

Short answer

\(\left(1 / x + 1 / y^2\right)^3 = 1 / x^3 + 3 / x^2 y^2 + 3 / x y^4 + 1 / y^6\)

Step-by-step solution

Expand using the Binomial Theorem

Use the Binomial Theorem to expand the binomial \(\left(1 / x + 1 / y^2\right)^3\). The general form of the Binomial Theorem is \( (a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k}b^k \) for any positive integer n. Here, a is \(1/x\), b is \(1/y^2\), and n is 3.

Calculate the Binomial Coefficients

Calculate the binomial coefficients using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) or by reading off Pascal's triangle. Since n is 3, the relevant row in Pascal's triangle is 1 3 3 1, which gives the binomial coefficients: 1, 3, 3, 1.

Apply the Binomial Coefficients

Apply the binomial coefficients to the expansion by multiplying the coefficients with the corresponding terms raised to the appropriate powers. This results in: \[\binom{3}{0}(\frac{1}{x})^{3-0}(\frac{1}{y^2})^0 + \binom{3}{1}(\frac{1}{x})^{3-1}(\frac{1}{y^2})^1 + \binom{3}{2}(\frac{1}{x})^{3-2}(\frac{1}{y^2})^2 + \binom{3}{3}(\frac{1}{x})^{3-3}(\frac{1}{y^2})^3\].

Key concepts

Binomial Coefficients

Understanding binomial coefficients is pivotal when studying the expansion of binomial expressions. Represented as \( \binom{n}{k} \) and pronounced 'n choose k', binomial coefficients determine the number of ways you can choose k items from a set of n items without regard to the order. In terms of the Binomial Theorem, they serve as multipliers for the terms of the expanded polynomial.

When expanding \( (1/x + 1/y^2)^3 \) using the Binomial Theorem, the binomial coefficients \( \binom{3}{k} \) for k ranging from 0 to 3 are critical for obtaining each term in the series. They are calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( ! \) denotes factorial. For our particular case, \( \binom{3}{0} \) = \( \binom{3}{3} \) = 1 and \( \binom{3}{1} \) = \( \binom{3}{2} \) = 3, as there is one way to select none or all items, and three ways to select one or two items from a set of three.

Pascal's Triangle

Pascal's triangle is a useful visual tool for quickly identifying the binomial coefficients needed for expanding binomials. This triangular array starts with a 1 at the top, and each subsequent row contains numbers that are the sum of the two directly above it. The nth row of Pascal's triangle corresponds to the coefficients of the expanded form of \( (a+b)^{n-1} \).

When we look at the fourth row (since the top row is considered row 0), we find the numbers 1, 3, 3, and 1, which we recognize as the binomial coefficients for our exercise, \( (1/x + 1/y^2)^3 \). This method serves as a quick reference and avoids the necessity of performing calculations to determine the coefficients.

Factorial Notation

To completely grasp binomial coefficients and, by extension, the Binomial Theorem, you must be comfortable with factorial notation. A factorial is denoted by the symbol \( ! \) and is the product of all positive integers from 1 up to the number. For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

A special case to remember is \( 0! \) which is defined as 1. Factorials are used in the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) for calculating binomial coefficients. They help in understanding the number of combinations of items and are heavily involved in computing probabilities and combinations in mathematics.

Polynomial Expansion

Finally, polynomial expansion is the process of expressing a polynomial raised to a power as a sum of terms. The Binomial Theorem provides a systematic method for expanding binomials of the form \( (a+b)^n \). Each term in the expanded polynomial is of the form \( \binom{n}{k} a^{n-k} b^k \), where \( n \) is the exponent, \( a \) and \( b \) are terms from the original binomial, and \( k \) takes on values from 0 through \( n \) inclusive.

In our example, \( (1/x + 1/y^2)^3 \) expands to \( 1/x^3 + 3/x^2y^2 + 3/xy^4 + 1/y^6 \) by multiplying each binomial coefficient with the corresponding powers of \( 1/x \) and \( 1/y^2 \). This results in a polynomial with four terms, which can be applied to various mathematical contexts from algebra to calculus and beyond.