Chapter 12: Q. 12.66. (page 1206)

Use Pascal’s 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

Step 1. Given information

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

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

According to Pascal's triangle,

${(a + b)}^{n} = a^{n} +_a^{n - 1} b^{1} +_a^{n - 2} b^{2} + \dots + b^{n}$

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

Short Answer

Step by step solution

Step 1. Given information

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

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