Problem 19 Expand. \(\left(x^{-2}+x^{2}\r... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 11: Problem 19

Expand. $\left(x^{-2}+x^{2}\right)^{4}$

Short Answer

Expert verified

The expansion is $x^{-8} + 4x^{-4} + 6 + 4x^{4} + x^{8}$.

Step by step solution

Apply the Binomial Theorem

The Binomial Theorem states \[ (a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \text{...} + \binom{n}{n} a^0 b^n \]Set $a = x^{-2}, b = x^2,$ and $n = 4$ in the Binomial Theorem.

Write out the binomial expansion terms

Use the expansion formula with $a = x^{-2}, b = x^2,$ and $n = 4$:\[ (x^{-2} + x^2)^4 = \binom{4}{0} (x^{-2})^4 (x^2)^0 + \binom{4}{1} (x^{-2})^3 (x^2)^1 + \binom{4}{2} (x^{-2})^2 (x^2)^2 + \binom{4}{3} (x^{-2})^1 (x^2)^3 + \binom{4}{4} (x^{-2})^0 (x^2)^4 \]

Calculate binomial coefficients and simplify each term

The binomial coefficients can be calculated as follows:\[ \binom{4}{0} = 1, \binom{4}{1} = 4, \binom{4}{2} = 6, \binom{4}{3} = 4, \binom{4}{4} = 1 \]Now simplify each term:\[ = 1 \times (x^{-2})^4 (x^2)^0 + 4 \times (x^{-2})^3 (x^2)^1 + 6 \times (x^{-2})^2 (x^2)^2 + 4 \times (x^{-2})^1 (x^2)^3 + 1 \times (x^{-2})^0 (x^2)^4 \]\[ = x^{-8} + 4x^{-6+2} + 6x^{-4+4} + 4x^{-2+6} + x^{0+8} \]\[ = x^{-8} + 4x^{-4} + 6x^{0} + 4x^{4} + x^{8} \]

Combine like terms if necessary

Combine the simplified expressions:\[ = x^{-8} + 4x^{-4} + 6 + 4x^{4} + x^{8} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expansion

The concept of expansion in mathematics often involves expressing an equation or expression in an extended form. When dealing with polynomials, the binomial theorem helps to simplify this expansion process. The binomial theorem states that for any binomial $a + b$ raised to the power of $n$, we can express it as the sum of terms involving binomial coefficients. This allows us to break down and systematically compute each term of the expansion.

Binomial Coefficients

Binomial coefficients play a critical role in the binomial expansion. They denote the number of ways to choose $k$ elements from a set of $n$ elements and are written as $\binom{n}{k}$. For instance, if you are expanding $(x^{-2} + x^2)^4$, you use the binomial coefficients $ \binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3},$ and $ \binom{4}{4} $. These coefficients are determined using the combination formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. Essentially, binomial coefficients help in determining the weights of each term in the expanded form.

Simplification

Simplification is the process of reducing a complex expression to its simplest form. Once you have used the binomial theorem to expand $ (x^{-2} + x^{2})^{4}$, you then need to simplify each resulting term. This includes calculating powers and combining like terms. In our example, after computing the power for each term, you get: $x^{-8} + 4x^{-4} + 6x^0 + 4x^4 + x^8$. Notice how the simplification step makes it easier to see and understand the resulting polynomial. Combining like terms, when required, ensures that the expression is as concise and readable as possible.

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91影视

Expand. \(\left(x^{-2}+x^{2}\right)^{4}\)

Short Answer

Step by step solution

Apply the Binomial Theorem

Write out the binomial expansion terms

Calculate binomial coefficients and simplify each term

Combine like terms if necessary

Key Concepts

Expansion

Binomial Coefficients

Simplification

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