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

$L e t x a n d a b e r e a l n u m b e r s . F o r a n y p o s i t i v e i n t e g e r n, w e h a v e {(x + a)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) x^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a x^{n - 1} + (\begin{matrix} n \\ 2 \end{matrix}) a^{2} x^{n - 2} + . . . . . . . + (\begin{matrix} n \\ j \end{matrix}) a^{j} x^{n - j} + . . . . . + (\begin{matrix} n \\ n \end{matrix}) a^{n} {(\sqrt{x} + \frac{3}{\sqrt{x}})}^{8} = (\begin{matrix} 8 \\ 0 \end{matrix}) {(\frac{3}{\sqrt{x}})}^{0} {(\sqrt{x})}^{8} + (\begin{matrix} 8 \\ 1 \end{matrix}) \times {(\frac{3}{\sqrt{x}})}^{1} {\times (\sqrt{x})}^{8 - 1} + (\begin{matrix} 8 \\ 2 \end{matrix}) \times {(\frac{3}{\sqrt{x}})}^{2} {\times (\sqrt{x})}^{8 - 2} + (\begin{matrix} 8 \\ 3 \end{matrix}) \times {(\frac{3}{\sqrt{x}})}^{3} {\times (\sqrt{x})}^{8 - 3} + (\begin{matrix} 8 \\ 4 \end{matrix}) \times {(\frac{3}{\sqrt{x}})}^{4} {\times (\sqrt{x})}^{8 - 4} + (\begin{matrix} 8 \\ 5 \end{matrix}) \times {(\frac{3}{\sqrt{x}})}^{5} {\times (\sqrt{x})}^{8 - 5} + (\begin{matrix} 8 \\ 6 \end{matrix}) \times {(\frac{3}{\sqrt{x}})}^{6} {\times (\sqrt{x})}^{8 - 6} + (\begin{matrix} 8 \\ 7 \end{matrix}) \times {(\frac{3}{\sqrt{x}})}^{7} {\times (\sqrt{x})}^{8 - 7} + (\begin{matrix} 8 \\ 8 \end{matrix}) \times {(\frac{3}{\sqrt{x}})}^{8} {\times (\sqrt{x})}^{8 - 8} = (\begin{matrix} 8 \\ 0 \end{matrix}) x^{4} + (\begin{matrix} 8 \\ 1 \end{matrix}) 3 x^{3} + (\begin{matrix} 8 \\ 2 \end{matrix}) 9 x^{2} + (\begin{matrix} 8 \\ 3 \end{matrix}) 27 x^{1} + (\begin{matrix} 8 \\ 4 \end{matrix}) 81 x^{0} + (\begin{matrix} 8 \\ 5 \end{matrix}) 243 x^{- 1} + (\begin{matrix} 8 \\ 6 \end{matrix}) 729 x^{- 2} + (\begin{matrix} 8 \\ 7 \end{matrix}) 2197 x^{- 3} + (\begin{matrix} 8 \\ 8 \end{matrix}) 6561 x^{- 4}$ 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!} \times 9 = \frac{8 \times 7 \times 6!}{2 \times 6!} \times 9 = 252

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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

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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

Geometry

Applied Mathematics

Discrete Mathematics

Probability and Statistics

Mechanics Maths

Logic and Functions

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 ${(\sqrt{x} + \frac{3}{\sqrt{x}})}^{8}$