Chapter 12: Q.38 (page 836)

use the Binomial Theorem to find the indicated coefficient or term.
The sixth term in the expansion of (3x +2)⁸

Short Answer

Expert verified

The sixth term is

Step by step solution

8Step 1 . Given information

Here (3x+2)⁸ is given

We have to find out the sixth term

. Expansion of (3x+2)8 using The Binomial Theorem

The binomial theorem states that

$L e t x a n d a b e r e a l n u m b e r s . F o r a n y p o s i t i v e i n t e g e r n, w e h a v e {(x + a)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) x^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a x^{n - 1} + (\begin{matrix} n \\ 2 \end{matrix}) a^{2} x^{n - 2} + . . . . . . . + (\begin{matrix} n \\ j \end{matrix}) a^{j} x^{n - j} + . . . . . + (\begin{matrix} n \\ n \end{matrix}) a^{n} t h e n b a s e d b o n t h i s t h e o r e m {(3 x + 2)}^{8} b e c o m e s (3 x + 2) 8 = (\begin{matrix} 8 \\ 0 \end{matrix}) {(3 x)}^{8} + (\begin{matrix} 8 \\ 1 \end{matrix}) \times {(2)}^{1} {\times (3 x)}^{8 - 1} + (\begin{matrix} 8 \\ 2 \end{matrix}) \times {(2)}^{2} \times {(3 x)}^{8 - 2} + (\begin{matrix} 8 \\ 3 \end{matrix}) \times {(2)}^{3} \times {(3 x)}^{8 - 3} + (\begin{matrix} 8 \\ 4 \end{matrix}) \times {(2)}^{4} \times {(3 x)}^{8 - 4} + (\begin{matrix} 8 \\ 5 \end{matrix}) \times {(2)}^{5} \times {(3 x)}^{8 - 5} + . . . . . + (\begin{matrix} 8 \\ 8 \end{matrix}) \times {(2)}^{9} = (\begin{matrix} 8 \\ 0 \end{matrix}) {(3 x)}^{8} + (\begin{matrix} 8 \\ 1 \end{matrix}) \times 2 \times {(3 x)}^{7} + (\begin{matrix} 8 \\ 2 \end{matrix}) \times 4 \times {(3 x)}^{6} + (\begin{matrix} 8 \\ 3 \end{matrix}) \times 8 \times {(3 x)}^{5} + (\begin{matrix} 8 \\ 4 \end{matrix}) \times 16 \times {(3 x)}^{4} + (\begin{matrix} 8 \\ 5 \end{matrix}) \times 32 \times {(3 x)}^{3} + . . . . + (\begin{matrix} 8 \\ 8 \end{matrix}) 256$

Step 3. Finding the sixth term

Based on question , sixth term from the expansion is

$(\begin{matrix} 8 \\ 5 \end{matrix}) \times 32 \times {(3 x)}^{3} = \frac{8!}{5! 3!} \times 32 \times 27 \times x^{3} = \frac{8 \times 7 \times 6 \times 5!}{3 \times 2 \times 1 \times 5!} \times 32 \times 27 \times x^{3} = 48384 x^{3}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

use the Binomial Theorem to find the indicated coefficient or term.
The sixth term in the expansion of (3x +2)⁸

Short Answer

Step by step solution

8Step 1 . Given information

. Expansion of (3x+2)8 using The Binomial Theorem

Step 3. Finding the sixth term

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

use the Binomial Theorem to find the indicated coefficient or term.The sixth term in the expansion of (3x +2)8

Short Answer

Step by step solution

8Step 1 . Given information

. Expansion of (3x+2)8 using The Binomial Theorem

Step 3. Finding the sixth term

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

use the Binomial Theorem to find the indicated coefficient or term.
The sixth term in the expansion of (3x +2)⁸