Chapter 12: Q.40 (page 836)

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

Short Answer

Expert verified

the coefficient of x⁰ is 495

Step by step solution

Step 1. Given information

Here ${(x - \frac{1}{x^{2}})}^{12}$ is given

We have to find out the coefficient of x⁰ is 495

Step 2. Description of expanding x-1x212

The Binomial theorem states that

$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 - \frac{1}{x^{2}})}^{12} = (\begin{matrix} 12 \\ 0 \end{matrix}) {(- \frac{1}{x^{2}})}^{0} {(x)}^{12} + (\begin{matrix} 12 \\ 1 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{1} {\times (x)}^{12 - 1} + (\begin{matrix} 12 \\ 2 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{2} {\times (x)}^{12 - 2} + (\begin{matrix} 12 \\ 3 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{3} {\times (x)}^{12 - 3} + (\begin{matrix} 12 \\ 4 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{4} {\times (x)}^{12 - 4} + (\begin{matrix} 12 \\ 5 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{5} {\times (x)}^{12 - 5} + (\begin{matrix} 12 \\ 6 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{6} {\times (x)}^{12 - 6} + (\begin{matrix} 12 \\ 7 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{7} {\times (x)}^{12 - 7} + (\begin{matrix} 12 \\ 8 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{8} {\times (x)}^{12 - 8} + (\begin{matrix} 12 \\ 9 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{9} {\times (x)}^{12 - 9} + (\begin{matrix} 12 \\ 10 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{10} {\times (x)}^{12 - 10} + (\begin{matrix} 12 \\ 11 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{11} {\times (x)}^{12 - 11} + (\begin{matrix} 12 \\ 12 \end{matrix}) \times {(- \frac{1}{x^{2}})}^{12} {\times (x)}^{12 - 12} = (\begin{matrix} 12 \\ 0 \end{matrix}) x^{12} - (\begin{matrix} 12 \\ 1 \end{matrix}) x^{9} + (\begin{matrix} 12 \\ 2 \end{matrix}) x^{6} - (\begin{matrix} 12 \\ 3 \end{matrix}) x^{3} + (\begin{matrix} 12 \\ 4 \end{matrix}) x^{0} - (\begin{matrix} 12 \\ 5 \end{matrix}) x^{- 3} + (\begin{matrix} 12 \\ 6 \end{matrix}) x^{- 6} - (\begin{matrix} 12 \\ 7 \end{matrix}) x^{- 9} + (\begin{matrix} 12 \\ 8 \end{matrix}) x^{- 12} - (\begin{matrix} 12 \\ 9 \end{matrix}) x^{- 15} + (\begin{matrix} 12 \\ 10 \end{matrix}) x^{- 18} - (\begin{matrix} 12 \\ 11 \end{matrix}) x^{- 21} + (\begin{matrix} 12 \\ 12 \end{matrix}) x^{- 24}$

Step 3. Finding the coefficient of x0

The term containing x⁰ from the expansion is

$(\begin{matrix} 12 \\ 4 \end{matrix}) x^{0,} h e n c e c o e f f i c i e n t o f x^{0} i s (\begin{matrix} 12 \\ 4 \end{matrix}) = \frac{12!}{4! 8!} = \frac{12 \times 11 \times 10 \times 9 \times 8!}{4 \times 3 \times 2 \times 1 \times 8!} = 495$

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Description of expanding x-1x212

Step 3. Finding the coefficient of x0

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Description of expanding x-1x212

Step 3. Finding the coefficient of x0

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Decision Maths

Mechanics Maths

Statistics

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 ${(x - \frac{1}{x^{2}})}^{12}$