Problem 23 Write the binomial expansion for... [FREE SOLUTION]

91影视

A Graphical Approach to College Algebra

John Hornsby,Margaret L. Lial,Gary Rockswold

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 8: Problem 23

Write the binomial expansion for each expression. $$(x+y)^{6}$$

Short Answer

Expert verified

$(x+y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6$.

Step by step solution

Understand the Binomial Theorem

The binomial theorem states that $(x+y)^n$ can be expanded as $ \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k} $. This formula will be used to find the expansion of $(x+y)^6$.

Identify the Values

In this case, $n = 6$. We need to calculate the terms for $k = 0$ to $k = 6$, i.e., seven terms in total.

Calculate Each Term

For each term in the expansion, calculate $\binom{6}{k} x^{6-k} y^{k}$. We'll perform this calculation iteratively from $k=0$ to $k=6$.

Calculate the Binomial Coefficients

The necessary binomial coefficients are: $\binom{6}{0} = 1$, $\binom{6}{1} = 6$, $\binom{6}{2} = 15$, $\binom{6}{3} = 20$, $\binom{6}{4} = 15$, $\binom{6}{5} = 6$, and $\binom{6}{6} = 1$.

Compute Each Term from Coefficients

Combine these coefficients with the corresponding power of $x$ and $y$:- For $k=0$: $\binom{6}{0} x^{6-0} y^{0} = x^6$- For $k=1$: $\binom{6}{1} x^{6-1} y^{1} = 6x^5y$- For $k=2$: $\binom{6}{2} x^{6-2} y^{2} = 15x^4y^2$- For $k=3$: $\binom{6}{3} x^{6-3} y^{3} = 20x^3y^3$- For $k=4$: $\binom{6}{4} x^{6-4} y^{4} = 15x^2y^4$- For $k=5$: $\binom{6}{5} x^{6-5} y^{5} = 6xy^5$- For $k=6$: $\binom{6}{6} x^{6-6} y^{6} = y^6$

Write the Complete Expression

Combine all the terms to write the complete expansion of $(x+y)^6$.The expansion is: \[x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\]

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Key Concepts

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

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to any power. When we see $(x+y)^n$, we can write it as a series of terms. Each term in this expansion involves combinations of $x$ and $y$. The general formula is given by:

\[ \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k} \]

Here, $n$ is the exponent, $x$ and $y$ are the variables, and $\binom{n}{k}$ is the binomial coefficient. This coefficient can be calculated using:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

This theorem is fundamental in expanding binomial expressions effectively without directly multiplying each term.

Combinatorics

Combinatorics comes into play when calculating the binomial coefficients. In $(x+y)^6$, there are several coefficients, such as $\binom{6}{0}, \binom{6}{1}, \binom{6}{2}$, and so on. These coefficients represent the number of ways to choose $k$ items from $n$ items without regard to order. Here's a quick guide:

$\binom{6}{0} = 1$: Choosing 0 items from 6 is just one way.
$\binom{6}{1} = 6$: Choosing 1 item from 6 can be done in 6 ways.
$\binom{6}{2} = 15$: Choosing 2 items can be done in 15 ways.
Continuing this, you find $\binom{6}{3} = 20$, $\binom{6}{4} = 15$, and so forth.

These numbers become the coefficients of the terms in the expansion. Understanding how to calculate these is crucial for polynomial expansions.

Polynomial Expansion

Expanding a binomial like $(x+y)^6$ involves creating a polynomial with several terms. Each term is a combination of powers of $x$ and $y$, multiplied by a binomial coefficient. The process looks like this:

Start with the highest power of $x$, decreasing as you move forward.
The power of $y$ starts at zero and increases.
Combine these with the respective coefficients.

For instance, the first term $x^6$ comes from the coefficient $\binom{6}{0}$, while the second term $6x^5y$ uses $\binom{6}{1}$. The complete expansion leads to:

\[ x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6 \]

This systematic approach helps break down the complexity into manageable parts.

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91影视

Write the binomial expansion for each expression. $$(x+y)^{6}$$

Short Answer

Step by step solution

Understand the Binomial Theorem

Identify the Values

Calculate Each Term

Calculate the Binomial Coefficients

Compute Each Term from Coefficients

Write the Complete Expression

Key Concepts

Binomial Theorem

Combinatorics

Polynomial Expansion

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