Q. 31 Expand the expression using the ... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q. 31 (page 839)

Expand the expression using the Binomial Theorem:
${(3 x - 4)}^{4} .$

Short Answer

Expert verified

The binomial expansion of ${(3 x - 4)}^{4}$ is $81 x^{4} - 432 x^{3} + 864 x^{2} - 768 x + 256$ .

Step by step solution

Given information:

Consider the given expression ${(3 x - 4)}^{4} .$

Rewrite the expression ${(3 x - 4)}^{4}$ as ${[3 x + (- 4)]}^{4}$ .

On comparing with the general form of binomial theorem ${(x + a)}^{n}$ , we get,

$x = 3 x, a = - 4, n = 4$ .

Required Formulas:

Formula of Binomial Theorem:

${(x + a)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) x^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a x^{n - 1} + . . . + (\begin{matrix} n \\ n \end{matrix}) a^{n}$

Pascal Triangle :

Substitute the values in the binomial expansion formula:

${[3 x + (- 4)]}^{4} = (\begin{matrix} 4 \\ 0 \end{matrix}) {(3 x)}^{4} + (\begin{matrix} 4 \\ 1 \end{matrix}) (- 4) {(3 x)}^{3} + (\begin{matrix} 4 \\ 2 \end{matrix}) {(- 4)}^{2} {(3 x)}^{2} + (\begin{matrix} 4 \\ 3 \end{matrix}) {(- 4)}^{3} (3 x) + (\begin{matrix} 4 \\ 4 \end{matrix}) {(- 4)}^{4}$

Simplification:

Here $n = 4$ , refer 4^th row of the Pascal triangle, role="math" localid="1647057469360" $[(\begin{matrix} 4 \\ 0 \end{matrix}) = 1, (\begin{matrix} 4 \\ 1 \end{matrix}) = 4, (\begin{matrix} 4 \\ 2 \end{matrix}) = 6, (\begin{matrix} 4 \\ 3 \end{matrix}) = 4, (\begin{matrix} 4 \\ 4 \end{matrix}) = 1]$

$鈭� {[3 x + (- 4)]}^{4} = [1 脳 {(3 x)}^{4}] + [4 脳 (- 4) {(3 x)}^{3}] + [6 脳 {(- 4)}^{2} {(3 x)}^{2}] + [4 脳 {(- 4)}^{3} (3 x)] + [1 脳 {(- 4)}^{4}]$

$= [81 x^{4}] + [(- 16) 脳 27 x^{3}] + [96 脳 9 x^{2}] + [(- 256) 脳 3 x] + [256] = 81 x^{4} - 432 x^{3} + 864 x^{2} - 768 x + 256$

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91影视

Expand the expression using the Binomial Theorem:
${(3 x - 4)}^{4} .$

Short Answer

Step by step solution

Given information:

Required Formulas:

Substitute the values in the binomial expansion formula:

Simplification:

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Expand the expression using the Binomial Theorem:3x-44.

Short Answer

Step by step solution

Given information:

Required Formulas:

Substitute the values in the binomial expansion formula:

Simplification:

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Expand the expression using the Binomial Theorem:
${(3 x - 4)}^{4} .$