Chapter 12: Q.34 (page 836)

use the Binomial Theorem to find the indicated coefficient or term
The coefficient of x²in the expansion of (2x -3)⁹

Short Answer

Expert verified

The coefficient of x² is -144

Step by step solution

Step 1. Given information

(2x-3)⁹ is given..

we have to find coefficient of the term of x²

. Expansion of (2x-3)9 using The Binomial Theorem

The binomial theorem states that

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

using this theorem we can expand the given expression, here a=1 and n=9

localid="1647339977970" ${(2 x - 3)}^{9} = (\begin{matrix} 9 \\ 0 \end{matrix}) {(2 x)}^{9} + (\begin{matrix} 9 \\ 1 \end{matrix}) (- 1) {(2 x)}^{9 - 1} + (\begin{matrix} 9 \\ 2 \end{matrix}) {(- 1)}^{2} {(2 x)}^{9 - 2} + . . . . . . . . . . . . + (\begin{matrix} 9 \\ 7 \end{matrix}) {(- 1)}^{7} {(2 x)}^{9 - 7} + (\begin{matrix} 9 \\ 8 \end{matrix}) {(- 1)}^{8} {(2 x)}^{9 - 8} + (\begin{matrix} 9 \\ 9 \end{matrix}) {(- 1)}^{9} = (\begin{matrix} 9 \\ 0 \end{matrix}) {(2 x)}^{9} - (\begin{matrix} 9 \\ 1 \end{matrix}) {(2 x)}^{8} + (\begin{matrix} 9 \\ 2 \end{matrix}) {(2 x)}^{7} + . . . . . . . . - (\begin{matrix} 9 \\ 7 \end{matrix}) {(2 x)}^{2} + (\begin{matrix} 9 \\ 8 \end{matrix}) {(2 x)}^{1} - (\begin{matrix} 9 \\ 9 \end{matrix})$

Step 3. Description of finding the coefficient of x2

The coefficient of x² from the expanded of the given expression is

$(\begin{matrix} 9 \\ 7 \end{matrix}) {(2)}^{2} {(- 1)}^{7}, w h i c h i s e q u a l t o \frac{9!}{2! 7!} \times 2^{2} \times {(- 1)}^{7} = - \frac{9 \times 8 \times 7!}{2 \times 1 \times 7!} \times 4 = - 144$

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use the Binomial Theorem to find the indicated coefficient or term
The coefficient of x²in the expansion of (2x -3)⁹

Short Answer

Step by step solution

Step 1. Given information

. Expansion of (2x-3)9 using The Binomial Theorem

Step 3. Description of finding the coefficient of x2

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use the Binomial Theorem to find the indicated coefficient or termThe coefficient of x2in the expansion of (2x -3)9

Short Answer

Step by step solution

Step 1. Given information

. Expansion of (2x-3)9 using The Binomial Theorem

Step 3. Description of finding the coefficient of x2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Statistics

Geometry

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

use the Binomial Theorem to find the indicated coefficient or term
The coefficient of x²in the expansion of (2x -3)⁹