Expand Using Pascal's Triangle (2x-1)^6

${(2 x - 1)}^{6}$

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

$1 - 5 - 10 - 10 - 5 - 1$

$1 - 6 - 15 - 20 - 15 - 6 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(2 x - 1)}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 6 - 15 - 20 - 15 - 6 - 1$ .

$1 a^{6} b^{0} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 a^{0} b^{6}$

Substitute the actual values of $a$ $2 x$ and $b$ $- 1$ into the expression.

$1 {(2 x)}^{6} {(- 1)}^{0} + 6 {(2 x)}^{5} {(- 1)}^{1} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Simplify each term.

Multiply ${(2 x)}^{6}$ by $1$ .

${(2 x)}^{6} {(- 1)}^{0} + 6 {(2 x)}^{5} {(- 1)}^{1} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Apply the product rule to $2 x$ .

$2^{6} x^{6} {(- 1)}^{0} + 6 {(2 x)}^{5} {(- 1)}^{1} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Raise $2$ to the power of $6$ .

$64 x^{6} {(- 1)}^{0} + 6 {(2 x)}^{5} {(- 1)}^{1} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Anything raised to $0$ is $1$ .

$64 x^{6} \cdot 1 + 6 {(2 x)}^{5} {(- 1)}^{1} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $64$ by $1$ .

$64 x^{6} + 6 {(2 x)}^{5} {(- 1)}^{1} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Apply the product rule to $2 x$ .

$64 x^{6} + 6 (2^{5} x^{5}) {(- 1)}^{1} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Raise $2$ to the power of $5$ .

$64 x^{6} + 6 (32 x^{5}) {(- 1)}^{1} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $32$ by $6$ .

$64 x^{6} + 192 x^{5} {(- 1)}^{1} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Evaluate the exponent.

$64 x^{6} + 192 x^{5} \cdot - 1 + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $- 1$ by $192$ .

$64 x^{6} - 192 x^{5} + 15 {(2 x)}^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Apply the product rule to $2 x$ .

$64 x^{6} - 192 x^{5} + 15 (2^{4} x^{4}) {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Raise $2$ to the power of $4$ .

$64 x^{6} - 192 x^{5} + 15 (16 x^{4}) {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $16$ by $15$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} {(- 1)}^{2} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Raise $- 1$ to the power of $2$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} \cdot 1 + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $240$ by $1$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} + 20 {(2 x)}^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Apply the product rule to $2 x$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} + 20 (2^{3} x^{3}) {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Raise $2$ to the power of $3$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} + 20 (8 x^{3}) {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $8$ by $20$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} + 160 x^{3} {(- 1)}^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Raise $- 1$ to the power of $3$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} + 160 x^{3} \cdot - 1 + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $- 1$ by $160$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 15 {(2 x)}^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Apply the product rule to $2 x$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 15 (2^{2} x^{2}) {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Raise $2$ to the power of $2$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 15 (4 x^{2}) {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $4$ by $15$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} {(- 1)}^{4} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Raise $- 1$ to the power of $4$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} \cdot 1 + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $60$ by $1$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} + 6 {(2 x)}^{1} {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Simplify.

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} + 6 (2 x) {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $2$ by $6$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} + 12 x {(- 1)}^{5} + 1 {(2 x)}^{0} {(- 1)}^{6}$

Raise $- 1$ to the power of $5$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} + 12 x \cdot - 1 + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply $- 1$ by $12$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} - 12 x + 1 {(2 x)}^{0} {(- 1)}^{6}$

Multiply ${(2 x)}^{0}$ by $1$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} - 12 x + {(2 x)}^{0} {(- 1)}^{6}$

Apply the product rule to $2 x$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} - 12 x + 2^{0} x^{0} {(- 1)}^{6}$

Anything raised to $0$ is $1$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} - 12 x + 1 x^{0} {(- 1)}^{6}$

Multiply $x^{0}$ by $1$ .

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} - 12 x + x^{0} {(- 1)}^{6}$

Anything raised to $0$ is $1$ .

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

Multiply ${(- 1)}^{6}$ by $1$ .

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

Raise $- 1$ to the power of $6$ .

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

Expand Using Pascal's Triangle (2x-1)^6

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