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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 12: Q. 12.66. (page 1206)

Use Pascal鈥檚 Triangle to expand ${(2 x - 1)}^{6}$

Short Answer

Expert verified

The expanded form of ${(2 x - 1)}^{6}$ is,

64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} - 12 x + 1

Step by step solution

01

Step 1. Given information

The given binomial is ${(2 x - 1)}^{6}$

02

Step 2. Use Pascal’s Triangle formula for binomial expansion.

According to Pascal's triangle,

${(a + b)}^{n} = a^{n} +_a^{n - 1} b^{1} +_a^{n - 2} b^{2} + 鈰� + b^{n}$

Use this formula,

${(2 x - 1)}^{6} = {(2 x)}^{6} + 6 {(2 x)}^{5} {(- 1)}^{1} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4 +} 6 (2 x) {(- 1)}^{5} + {(- 1)}^{6} = 64 x^{6} + 6 (32 x^{5}) (- 1) + 15 (16 x^{4}) (1) + 20 (8 x^{3}) (- 1) + 15 (4 x^{2}) (1) + 6 (2 x) (- 1) + 1 = 64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} - 12 x + 1$

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Most popular questions from this chapter

In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.

$1, 2, 4, 8, 16, 32, 鈥�$

Write the sum using summation notation: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} .$

In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.

$a_{1} = 4$ and $r = 3$

The expansion is $y^{5} + 10 y^{4} + 40 y^{3} + 80 y^{2} + 80 y + 32$

Find the sum of the first $20$ terms of the geometric sequence $3, 6, 12, 24, 48, 96, 鈥�$ .

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