Chapter 12: Q.39 (page 836)

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x⁰ in the expansion of ${(x^{2} + \frac{1}{x})}^{12}$

Short Answer

Expert verified

the coefficient of x ⁰is 495

Step by step solution

Step 1. Given information

${(x^{2} + \frac{1}{x})}^{12}$ is given.

We have to find out the term x⁰

Step 2. Description of expanding (x2+1x)12

By using the Binomial theorem we can expand

$L e t x a n d a b e r e a l n u m b e r s . F o r a n y p o s i t i v e i n t e g e r n, w e h a v e {(x + a)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) x^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a x^{n - 1} + (\begin{matrix} n \\ 2 \end{matrix}) a^{2} x^{n - 2} + . . . . . . . + (\begin{matrix} n \\ j \end{matrix}) a^{j} x^{n - j} + . . . . . + (\begin{matrix} n \\ n \end{matrix}) a^{n} b a s e d o n t h i s {(x^{2} + \frac{1}{x})}^{12} b e c o m e s {(x^{2} + \frac{1}{x})}^{12} = (\begin{matrix} 12 \\ 0 \end{matrix}) {(\frac{1}{x})}^{0} (x$ ²)12+121×(1x)1×(x2)12-1+122×(1x)2×(x2)12-2+123×(1x)3×(x2)12-3+124×(1x)4×(x2)12-4+125×(1x)5×(x2)12-5+126×(1x)6×(x2)12-6+127×(1x)7×(x2)12-7+128×(1x)8×(x2)12-8+129×(1x)9×(x2)12-9+1210×(1x)10×(x2)12-10+1211×(1x)11×(x2)12-11+1212×(1x)12×(x2)12-12=120x24+121x21+122x18+123x15+124x12+125x9+126x6+127x3+128x0+129x-3+1210x-6+1211x-9+1212x-12

Step 3. Finding the coefficient of x0

The term containing x⁰ from the expansion is $(\begin{matrix} 12 \\ 8 \end{matrix}) x^{0}, t h e n t h e c o e f f i c i e n t o f x^{0} = (\begin{matrix} 12 \\ 8 \end{matrix}) = \frac{12!}{8! 4!} = \frac{12 \times 11 \times 10 \times 9 \times 8!}{4 \times 3 \times 2 \times 1 \times 8!} = 495$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x⁰ in the expansion of ${(x^{2} + \frac{1}{x})}^{12}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Description of expanding (x2+1x)12

Step 3. Finding the coefficient of x0

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Pure Maths

Logic and Functions

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x0 in the expansion of(x2+1x)12

Short Answer

Step by step solution

Step 1. Given information

Step 2. Description of expanding (x2+1x)12

Step 3. Finding the coefficient of x0

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Pure Maths

Logic and Functions

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x⁰ in the expansion of ${(x^{2} + \frac{1}{x})}^{12}$