Chapter 12: Q.32 (page 836)

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x³in the expansion of (2x+1)¹²

Short Answer

Expert verified

The coefficient $x^{3} i s$ 1760

Step by step solution

. Given information

here (2x+1)¹² is given

we need to find the coefficient of x³

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

According to the binomial theorem

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

Then the expansion of (2x+1)¹²becomes

${(2 x + 1)}^{12} = (\begin{matrix} 12 \\ 0 \end{matrix}) {(2 x)}^{12} + (\begin{matrix} 12 \\ 1 \end{matrix}) (1) {(2 x)}^{12 - 1} + . . . . . . . . . . . + (\begin{matrix} 12 \\ 9 \end{matrix}) {(1)}^{9} {(2 x)}^{12 - 9} + (\begin{matrix} 12 \\ 10 \end{matrix}) {(1)}^{10} {(2 x)}^{12 - 10} + (\begin{matrix} 12 \\ 11 \end{matrix}) {(1)}^{11} {(2 x)}^{12 - 11} + (\begin{matrix} 12 \\ 12 \end{matrix}) 1^{12} = (\begin{matrix} 12 \\ 0 \end{matrix}) {(2 x)}^{12} + (\begin{matrix} 12 \\ 1 \end{matrix}) {(2 x)}^{11} + . . . . . . . + (\begin{matrix} 12 \\ 9 \end{matrix}) {(2 x)}^{3} + (\begin{matrix} 12 \\ 10 \end{matrix}) {(2 x)}^{2} + (\begin{matrix} 12 \\ 11 \end{matrix}) {(2 x)}^{1} + (\begin{matrix} 12 \\ 12 \end{matrix})$

Step 3. Description of finding the coefficient of x3

According to the question, the coefficient of x³ from the expanded term is

$(\begin{matrix} 12 \\ 9 \end{matrix}) {(2)}^{3} {(1)}^{9}, w h i c h i s e q u a l t o \frac{12!}{3! 9!} \times 2^{3} \times 1 = \frac{12 \times 11 \times 10 \times 9!}{3 \times 2 \times 1 \times 9!} \times 8 = 1760, t h i s i s c o e . o f x^{3}$

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use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x³in 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 x3

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use the Binomial Theorem to find the indicated coefficient or term.The coefficient of x3in 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 x3

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Statistics

Mechanics Maths

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x³in the expansion of (2x+1)¹²