Q.42 use the Binomial Theorem to find... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q.42 (page 836)

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x² in the expansion of ${(\sqrt{x} + \frac{3}{\sqrt{x}})}^{8}$

Short Answer

Expert verified

The coefficient of x² is 252

Step by step solution

Step 1. Given information

In this question ${(\sqrt{x} + \frac{3}{\sqrt{x}})}^{8}$ is given.

We have to find out thecoefficient of x² in the expansion

Step 2. Description of expansion of x+3x8

$L e t x a n d a b e r e a l n u m b e r s . F o r a n y p o s i t i v e i n t e g e r n, w e h a v e {(x + a)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) x^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a x^{n - 1} + (\begin{matrix} n \\ 2 \end{matrix}) a^{2} x^{n - 2} + . . . . . . . + (\begin{matrix} n \\ j \end{matrix}) a^{j} x^{n - j} + . . . . . + (\begin{matrix} n \\ n \end{matrix}) a^{n} {(\sqrt{x} + \frac{3}{\sqrt{x}})}^{8} = (\begin{matrix} 8 \\ 0 \end{matrix}) {(\frac{3}{\sqrt{x}})}^{0} {(\sqrt{x})}^{8} + (\begin{matrix} 8 \\ 1 \end{matrix}) 脳 {(\frac{3}{\sqrt{x}})}^{1} {脳 (\sqrt{x})}^{8 - 1} + (\begin{matrix} 8 \\ 2 \end{matrix}) 脳 {(\frac{3}{\sqrt{x}})}^{2} {脳 (\sqrt{x})}^{8 - 2} + (\begin{matrix} 8 \\ 3 \end{matrix}) 脳 {(\frac{3}{\sqrt{x}})}^{3} {脳 (\sqrt{x})}^{8 - 3} + (\begin{matrix} 8 \\ 4 \end{matrix}) 脳 {(\frac{3}{\sqrt{x}})}^{4} {脳 (\sqrt{x})}^{8 - 4} + (\begin{matrix} 8 \\ 5 \end{matrix}) 脳 {(\frac{3}{\sqrt{x}})}^{5} {脳 (\sqrt{x})}^{8 - 5} + (\begin{matrix} 8 \\ 6 \end{matrix}) 脳 {(\frac{3}{\sqrt{x}})}^{6} {脳 (\sqrt{x})}^{8 - 6} + (\begin{matrix} 8 \\ 7 \end{matrix}) 脳 {(\frac{3}{\sqrt{x}})}^{7} {脳 (\sqrt{x})}^{8 - 7} + (\begin{matrix} 8 \\ 8 \end{matrix}) 脳 {(\frac{3}{\sqrt{x}})}^{8} {脳 (\sqrt{x})}^{8 - 8} = (\begin{matrix} 8 \\ 0 \end{matrix}) x^{4} + (\begin{matrix} 8 \\ 1 \end{matrix}) 3 x^{3} + (\begin{matrix} 8 \\ 2 \end{matrix}) 9 x^{2} + (\begin{matrix} 8 \\ 3 \end{matrix}) 27 x^{1} + (\begin{matrix} 8 \\ 4 \end{matrix}) 81 x^{0} + (\begin{matrix} 8 \\ 5 \end{matrix}) 243 x^{- 1} + (\begin{matrix} 8 \\ 6 \end{matrix}) 729 x^{- 2} + (\begin{matrix} 8 \\ 7 \end{matrix}) 2197 x^{- 3} + (\begin{matrix} 8 \\ 8 \end{matrix}) 6561 x^{- 4}$ We can expand the given expression using The Binomial theorem, It states that

Step 3. Finding the coefficient of x2

From the expansion of the given expression, coefficient of x²is

(\begin{matrix} 8 \\ 2 \end{matrix}) 9 = \frac{8!}{6! 2!} 脳 9 = \frac{8 脳 7 脳 6!}{2 脳 6!} 脳 9 = 252

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91影视

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x² in the expansion of ${(\sqrt{x} + \frac{3}{\sqrt{x}})}^{8}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Description of expansion of x+3x8

Step 3. Finding the coefficient of x2

One App. One Place for Learning.

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91影视

use the Binomial Theorem to find the indicated coefficient or term.The coefficient of x2 in the expansion ofx+3x8

Short Answer

Step by step solution

Step 1. Given information

Step 2. Description of expansion of x+3x8

Step 3. Finding the coefficient of x2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Mechanics Maths

Theoretical and Mathematical Physics

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x² in the expansion of ${(\sqrt{x} + \frac{3}{\sqrt{x}})}^{8}$