Chapter 12: Q.37 (page 836)

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

Short Answer

Expert verified

The third term is $314928 x^{7}$

Step by step solution

. Given information

In this question (3x-2)⁹ is given.

We have to find out the third term

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

According to the binomial theorem

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

So given expression becomes

${(3 x - 2)}^{9} = (\begin{matrix} 9 \\ 0 \end{matrix}) {(3 x)}^{9} + (\begin{matrix} 9 \\ 1 \end{matrix}) \times {(- 2)}^{1} {\times (3 x)}^{9 - 1} + (\begin{matrix} 9 \\ 2 \end{matrix}) \times {(- 2)}^{2} \times {(3 x)}^{9 - 2} + . . . . . + (\begin{matrix} 9 \\ 8 \end{matrix}) \times {(- 2)}^{8} {\times (3 x)}^{9 - 8} + (\begin{matrix} 9 \\ 9 \end{matrix}) \times {(- 2)}^{9} = (\begin{matrix} 9 \\ 0 \end{matrix}) {(3 x)}^{9} - (\begin{matrix} 9 \\ 1 \end{matrix}) \times 2 \times {(3 x)}^{8} + (\begin{matrix} 9 \\ 2 \end{matrix}) \times 4 \times {(3 x)}^{7} + . . . . + (\begin{matrix} 9 \\ 8 \end{matrix}) 256 (3 x) - (\begin{matrix} 9 \\ 9 \end{matrix}) 512$

Step 3. Description of finding the third term

The third term from the expanded form of the given expression is

$(\begin{matrix} 9 \\ 2 \end{matrix}) \times 4 \times {(3 x)}^{7} = \frac{9!}{2! 7!} \times 4 \times 3^{7} \times x^{7} = \frac{9 \times 8 \times 7!}{1 \times 2 \times 7!} \times 4 \times 3^{7} \times x^{7} = 314928 x^{7}$

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

Short Answer

Step by step solution

. Given information

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

Step 3. Description of finding the third term

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

Short Answer

Step by step solution

. Given information

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

Step 3. Description of finding the third term

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Pure Maths

Applied Mathematics

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

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