Problem 28 Use the binomial theorem to expa... [FREE SOLUTION]

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Precalculus: Functions and Graphs

Earl W. Swokowski, Jeffrey A. Cole

$Math Studyset 91影视 Explanations$ Math

12 Edition

Chapter 9: Problem 28

Use the binomial theorem to expand and simplify. $$\left(\frac{1}{2} x+y^{3}\right)^{4}$$

Short Answer

Expert verified

$\left(\frac{1}{2}x + y^3\right)^4 = \frac{1}{16}x^4 + \frac{1}{2}x^3y^3 + \frac{3}{2}x^2y^6 + 2xy^9 + y^{12}$.

Step by step solution

Identify the Binomial Formula

The binomial theorem states that \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]. In this exercise, the binomial expression is $(\frac{1}{2}x + y^3)^4$, where $a = \frac{1}{2}x$, $b = y^3$, and $n = 4$.

Calculate Binomial Coefficients

The binomial coefficients are defined as $ \binom{n}{k} = \frac{n!}{k!(n-k)!} $. For $ n = 4 $, the coefficients are:- $ \binom{4}{0} = 1 $- $ \binom{4}{1} = 4 $- $ \binom{4}{2} = 6 $- $ \binom{4}{3} = 4 $- $ \binom{4}{4} = 1 $

Write and Simplify Each Term

Using the binomial theorem, expand each term separately:- For $ k = 0 $: $\binom{4}{0}(\frac{1}{2}x)^{4-0}(y^3)^{0} = 1 \cdot (\frac{1}{2}x)^4 = \frac{1}{16}x^4$- For $ k = 1 $: $\binom{4}{1}(\frac{1}{2}x)^{4-1}(y^3)^{1} = 4 \cdot (\frac{1}{2}x)^3 \cdot y^3 = \frac{1}{2}x^3y^3$- For $ k = 2 $: $\binom{4}{2}(\frac{1}{2}x)^{4-2}(y^3)^{2} = 6 \cdot (\frac{1}{2}x)^2 \cdot y^6 = \frac{3}{2}x^2y^6$- For $ k = 3 $: $\binom{4}{3}(\frac{1}{2}x)^{4-3}(y^3)^{3} = 4 \cdot (\frac{1}{2}x) \, y^9 = 2xy^9$- For $ k = 4 $: $\binom{4}{4}(\frac{1}{2}x)^{4-4}(y^3)^{4} = 1 \cdot y^{12} = y^{12}$

Combine All Terms

Combine the results from Step 3 to arrive at the final expanded form of the expression:\[ \left(\frac{1}{2}x + y^3\right)^4 = \frac{1}{16}x^4 + \frac{1}{2}x^3y^3 + \frac{3}{2}x^2y^6 + 2xy^9 + y^{12} \]

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Key Concepts

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

Binomial Coefficients

Binomial coefficients play a crucial role in expanding expressions using the binomial theorem. These coefficients come from the patterns in expanding binomials, like \[ (a + b)^n \]. Each coefficient can be found using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \], where $ n! $ ("n factorial") is the product of all positive integers up to $ n $.
To better understand, consider the binomial expression $(a + b)^4$. The coefficients for this expansion are determined as follows:

$ \binom{4}{0} = 1 $
$ \binom{4}{1} = 4 $
$ \binom{4}{2} = 6 $
$ \binom{4}{3} = 4 $
$ \binom{4}{4} = 1 $

These numbers are the coefficients that correspond to each term in the expanded binomial. They're crucial since they tell us how many times each term needs to be multiplied within the expression.

Polynomial Expansion

Polynomial expansion involves expressing a binomial raised to a power as a polynomial. This is achieved using the binomial theorem, which breaks down the expression $ (a + b)^n $ into a sum of $ n+1 $ terms. Each term follows the pattern $ \binom{n}{k} a^{n-k} b^k $.
For example, the expression $(\frac{1}{2}x + y^3)^4$ can be expanded into a polynomial by following these steps:

Identify the values of $ a $, $ b $, and $ n $.
Calculate the binomial coefficients for each $ k $ from $ 0 $ to $ n $.
Use each coefficient to write the corresponding term.
Simplify each term if needed, applying powers and distributing any constants.

The expanded form provides a clear and simplified polynomial expression, making it easier to perform arithmetic operations or apply further algebraic methods.

Pascal's Triangle

Pascal's Triangle is a fascinating and practical tool for finding binomial coefficients quickly. Each row in the triangle corresponds to the coefficients for expanding $ (a + b)^n $. You start with a simple '1' at the top (the zeroth row).
Each subsequent number in the triangle is the sum of the two numbers directly above it in the previous row.
Here's a look at some starting rows of Pascal's Triangle:

Row 0: 1
Row 1: 1, 1
Row 2: 1, 2, 1
Row 3: 1, 3, 3, 1
Row 4: 1, 4, 6, 4, 1

For the expression $(\frac{1}{2}x + y^3)^4$, the coefficients $ 1, 4, 6, 4, 1 $ can be directly taken from the fourth row of Pascal's Triangle. This makes calculations much easier, especially for larger $ n $ values, since manual calculations of factorials might be cumbersome.

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91影视

Use the binomial theorem to expand and simplify. $$\left(\frac{1}{2} x+y^{3}\right)^{4}$$

Short Answer

Step by step solution

Identify the Binomial Formula

Calculate Binomial Coefficients

Write and Simplify Each Term

Combine All Terms

Key Concepts

Binomial Coefficients

Polynomial Expansion

Pascal's Triangle

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