Chapter 12: Q.33 (page 836)

use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x⁷ in the expansion of (2x +3)⁹

Short Answer

Expert verified

The coefficient of x⁷ is 4608

Step by step solution

. Given information

Here (2x+3)⁹ is given.

we have to find out the coefficient of the term x⁷

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

According to the binomial theorem

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

Based on this we can expand, here a=1 and n=9

localid="1647339902877" ${(2 x + 3)}^{9} = (\begin{matrix} 9 \\ 0 \end{matrix}) {(2 x)}^{9} + (\begin{matrix} 9 \\ 1 \end{matrix}) (1) {(2 x)}^{9 - 1} + (\begin{matrix} 9 \\ 2 \end{matrix}) {(1)}^{2} {(2 x)}^{9 - 2} + (\begin{matrix} 9 \\ 3 \end{matrix}) {(1)}^{3} {(2 x)}^{9 - 3} + . . . . . . + (\begin{matrix} 9 \\ 8 \end{matrix}) {(1)}^{8} {(2 x)}^{9 - 8} + (\begin{matrix} 9 \\ 9 \end{matrix}) 1^{9} = (\begin{matrix} 9 \\ 0 \end{matrix}) {(2 x)}^{9} + (\begin{matrix} 9 \\ 1 \end{matrix}) {(2 x)}^{8} + (\begin{matrix} 9 \\ 2 \end{matrix}) {(2 x)}^{7} + (\begin{matrix} 9 \\ 3 \end{matrix}) {(2 x)}^{6} + . . . . . . + (\begin{matrix} 9 \\ 8 \end{matrix}) {(2 x)}^{1} + (\begin{matrix} 9 \\ 9 \end{matrix})$

Step 3. Description of finding the coefficient of x7

From the expanded form of (2x+3)⁹ we get a coefficient of x⁷

then the coefficient of x⁷ is

$(\begin{matrix} 9 \\ 2 \end{matrix}) {(2)}^{7} {(1)}^{2}, w h i c h i s e q u a l t o \frac{9!}{2! 7!} \times 2^{7} = \frac{9 \times 8 \times 7!}{2 \times 1 \times 7!} \times 128 = 4608$

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use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of x⁷ in the expansion of (2x +3)⁹

Short Answer

Step by step solution

. Given information

. Expansion of (2x+3)9 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 +3)9

Short Answer

Step by step solution

. Given information

. Expansion of (2x+3)9 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

Decision Maths

Discrete Mathematics

Pure Maths

Applied Mathematics

Theoretical and Mathematical Physics

Logic and Functions

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 +3)⁹