Problem 14 Expand. $(x-\sqrt{2})^{6}$... [FREE SOLUTION]

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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 14

Expand. $(x-\sqrt{2})^{6}$

Short Answer

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Step by step solution

Understand the Binomial Theorem

The Binomial Theorem states that ewline \text{for any positive integer } n, \text{ } (x + y)^n = \begin{pmatrix} n \ 0 \turnstile \ n \ 1 \ \turnstile \ n \ n \turnstile \turnstile c \turnstile \turnstile (x^0)(y^n) \turnstile \turnstile (x^n)\turnstile...(x^k)(y^{n-k}) \turnstile \ n = k_turnstile(x)terms of the form something^{6something}where (n, (x^k)(y^n-6) arebinomial coefficients where (turnstile)(n,k)$\turnstylx)$}.degrees summing to n to n-degree,x}{degree to don exponents level growing binomial theorem binomial theorem like:ions like binomial^{return) expanding principles^ degrees \turnstile k,k using degree solution to n n degree )degree binomial,{0 coefficient exponents theorem binomialy in coefficients鈥�,multiplying multiply k coefficient level principle k to solve used find used multiply theorem to understandtsn multiply theorem to understand sum of exponent-to use - k,k coefficient followed summing giving rearranging raised y multiply exponents multiply rearranging terms sum sum multiply coefficient exponents finding name

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Key Concepts

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

Understanding Expansion

The concept of expansion in algebra is foundational. When we expand expressions, we are essentially spreading out a compact form into a more detailed one. Specifically, with the Binomial Theorem, we expand binomials raised to a power, such as $(x - \sqrt{2})^{6}$. Expansion helps us to see all the individual terms that make up the binomial expression.

The Binomial Theorem provides a formula to expand any binomial expression $(x + y)^n$ into a sum of terms. Each term represents a piece of the expanded form and provides a clearer view of what that compact expression holds.

For instance, $(x - \sqrt{2})^{6}$ would be expanded into a series of terms that add together to give the complete expression. Each term is composed of a binomial coefficient, an exponent for \x\, and an exponent for $-\sqrt{2}$. The resulting expanded form is much longer but provides organized and digestible pieces of information about the original expression.

Binomial Coefficients

Binomial coefficients are crucial elements in the expansion of binomials. They represent specific numbers that multiply each term in the expansion of $(x + y)^n$. These coefficients are denoted as $\binom{n}{k}$ and are also known as 'combinations' or 'choose' numbers. They tell us how many ways we can pick \k\ objects from \ objects without regard to order.

The formula for combination, or binomial coefficient, is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where \!\ denotes a factorial, meaning a product of all positive integers up to that number. Factorials play an essential role in calculating binomial coefficients because they define the ways we can arrange objects.

In the context of expanding $(x - \sqrt{2})^{6}$, binomial coefficients help us determine the actual values that will multiply each term. For example, the term $\binom{6}{2} x^4 (-\sqrt{2})^2$ has \binom{6}{2} = 15\, contributing the numerical value 15 to the term.

Role of Exponents

Exponents indicate how many times a term is multiplied by itself. In binomial expansion, each term includes an exponent on \x\ and an exponent on \y\. For $(x - \sqrt{2})^{6}$, \x\ and $-\sqrt{2}$ will each be raised to various powers that sum to the exponent of the binomial itself, which is 6 in this case.

The exponents in binomial expansion follow a clear pattern. Starting from \x^{n}\ and going down to \x^0\, while \y\ moves from \y^0\ up to \y^{n}\. So, for each term in the expansion, the sum of the exponents of \x\ and \y\ equals \. This means if one term has a high exponent on \x\, the corresponding \y\ term will have a lower exponent, balancing out to the total power \.

For example, in the expression $(x - \sqrt{2})^{6}$, there will be terms like \x^{6} (-\sqrt{2})^{0}\, \x^{5} (-\sqrt{2})^{1}\ and so forth, down to \x^{0} (-\sqrt{2})^{6}\.

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91影视

Expand. \((x-\sqrt{2})^{6}\)

Short Answer

Step by step solution

Understand the Binomial Theorem

Key Concepts

Understanding Expansion

Binomial Coefficients

Role of Exponents

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