Chapter 12: Q.31 (page 836)

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x⁷ in the expansion of (2x - 1)¹²

Short Answer

Expert verified

The coefficient of $x^{7} i n {(2 x - 1)}^{12} i s - 101376$

Step by step solution

. Given information

${(2 x - 7)}^{12}$ is given.

We have to find the coefficient of term x⁷

step 2 . Expansion of (2x-1)12 using The Binomial Theorem

The binomial theorem state that

$L e t x a n d a b e r e a l n u m b e r s . F o r a n y p o s i t i v e i n t e g e r n, w e h a v e {(x + a)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) x^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a x^{n - 1} + (\begin{matrix} n \\ 2 \end{matrix}) a^{2} x^{n - 2} + . . . . . . . + (\begin{matrix} n \\ j \end{matrix}) a^{j} x^{n - j} + . . . . . + (\begin{matrix} n \\ n \end{matrix}) a^{n}$

using this we can expand (2x-12)¹² here a= -1 and n=12 ,then, we get ${(2 x - 1)}^{12} = (\begin{matrix} 12 \\ 0 \end{matrix}) {(2 x)}^{12} + (\begin{matrix} 12 \\ 1 \end{matrix}) (- 1) {(2 x)}^{12 - 1} + (\begin{matrix} 12 \\ 2 \end{matrix}) {(- 1)}^{2} {(2 x)}^{12 - 2} + (\begin{matrix} 12 \\ 3 \end{matrix}) {(- 1)}^{3} {(2 x)}^{12 - 3} + (\begin{matrix} 12 \\ 4 \end{matrix}) {(- 1)}^{4} {(2 x)}^{12 - 4} + (\begin{matrix} 12 \\ 5 \end{matrix}) {(- 1)}^{5} {(2 x)}^{12 - 5} + . . . . . . + (\begin{matrix} 12 \\ 11 \end{matrix}) {(- 1)}^{11} {(2 x)}^{12 - 11} + (\begin{matrix} 12 \\ 12 \end{matrix}) 1^{12} = (\begin{matrix} 12 \\ 0 \end{matrix}) {(2 x)}^{12} - (\begin{matrix} 12 \\ 1 \end{matrix}) {(2 x)}^{11} + (\begin{matrix} 12 \\ 2 \end{matrix}) {(2 x)}^{10} - (\begin{matrix} 12 \\ 3 \end{matrix}) {(2 x)}^{9} + (\begin{matrix} 12 \\ 4 \end{matrix}) {(2 x)}^{8} - (\begin{matrix} 12 \\ 5 \end{matrix}) {(2 x)}^{7} + . . . . . . - (\begin{matrix} 12 \\ 11 \end{matrix}) {(2 x)}^{1} + (\begin{matrix} 12 \\ 12 \end{matrix})$

Step 3. Description of finding the coefficient of x7

According to the question, the coefficient of x⁷ from the expanded term is

$(\begin{matrix} 12 \\ 5 \end{matrix}) {(2)}^{7} {(- 1)}^{5} = \frac{12!}{5! 7!} \times 2^{7} \times - 1 = - \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7!}{5 \times 4 \times 3 \times 2 \times 1 \times 7!} \times 128 = - 101376, t h i s i s t h e c o e . o f x^{7}$

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use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x⁷ in the expansion of (2x - 1)¹²

Short Answer

Step by step solution

. Given information

step 2 . Expansion of (2x-1)12 using The Binomial Theorem

Step 3. Description of finding the coefficient of x7

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use the Binomial Theorem to find the indicated coefficient or term.The coefficient of x7 in the expansion of (2x - 1)12

Short Answer

Step by step solution

. Given information

step 2 . Expansion of (2x-1)12 using The Binomial Theorem

Step 3. Description of finding the coefficient of x7

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x⁷ in the expansion of (2x - 1)¹²