Expand Using Pascal's Triangle (2-x)^5

${(2 - x)}^{5}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

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 ${(2 - x)}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

Step 2

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

$1 a^{5} b^{0} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 a^{0} b^{5}$

Step 3

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

$1 {(2)}^{5} {(- x)}^{0} + 5 {(2)}^{4} {(- x)}^{1} + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Step 4

Simplify each term.

Multiply ${(2)}^{5}$ by $1$ .

${(2)}^{5} {(- x)}^{0} + 5 {(2)}^{4} {(- x)}^{1} + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

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

$32 {(- x)}^{0} + 5 {(2)}^{4} {(- x)}^{1} + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Apply the product rule to $- x$ .

$32 ({(- 1)}^{0} x^{0}) + 5 {(2)}^{4} {(- x)}^{1} + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Anything raised to $0$ is $1$ .

$32 (1 x^{0}) + 5 {(2)}^{4} {(- x)}^{1} + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

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

$32 x^{0} + 5 {(2)}^{4} {(- x)}^{1} + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Anything raised to $0$ is $1$ .

$32 \cdot 1 + 5 {(2)}^{4} {(- x)}^{1} + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Multiply $32$ by $1$ .

$32 + 5 {(2)}^{4} {(- x)}^{1} + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

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

$32 + 5 \cdot 16 {(- x)}^{1} + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Multiply $5$ by $16$ .

$32 + 80 {(- x)}^{1} + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Simplify.

$32 + 80 (- x) + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Multiply $- 1$ by $80$ .

$32 - 80 x + 10 {(2)}^{3} {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

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

$32 - 80 x + 10 \cdot 8 {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Multiply $10$ by $8$ .

$32 - 80 x + 80 {(- x)}^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Apply the product rule to $- x$ .

$32 - 80 x + 80 ({(- 1)}^{2} x^{2}) + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

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

$32 - 80 x + 80 (1 x^{2}) + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Multiply $x^{2}$ by $1$ .

$32 - 80 x + 80 x^{2} + 10 {(2)}^{2} {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

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

$32 - 80 x + 80 x^{2} + 10 \cdot 4 {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Multiply $10$ by $4$ .

$32 - 80 x + 80 x^{2} + 40 {(- x)}^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Apply the product rule to $- x$ .

$32 - 80 x + 80 x^{2} + 40 ({(- 1)}^{3} x^{3}) + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

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

$32 - 80 x + 80 x^{2} + 40 (- x^{3}) + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Multiply $- 1$ by $40$ .

$32 - 80 x + 80 x^{2} - 40 x^{3} + 5 {(2)}^{1} {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Evaluate the exponent.

$32 - 80 x + 80 x^{2} - 40 x^{3} + 5 \cdot 2 {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Multiply $5$ by $2$ .

$32 - 80 x + 80 x^{2} - 40 x^{3} + 10 {(- x)}^{4} + 1 {(2)}^{0} {(- x)}^{5}$

Apply the product rule to $- x$ .

$32 - 80 x + 80 x^{2} - 40 x^{3} + 10 ({(- 1)}^{4} x^{4}) + 1 {(2)}^{0} {(- x)}^{5}$

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

$32 - 80 x + 80 x^{2} - 40 x^{3} + 10 (1 x^{4}) + 1 {(2)}^{0} {(- x)}^{5}$

Multiply $x^{4}$ by $1$ .

$32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} + 1 {(2)}^{0} {(- x)}^{5}$

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

$32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} + {(2)}^{0} {(- x)}^{5}$

Anything raised to $0$ is $1$ .

$32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} + 1 {(- x)}^{5}$

Multiply ${(- x)}^{5}$ by $1$ .

$32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} + {(- x)}^{5}$

Apply the product rule to $- x$ .

$32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} + {(- 1)}^{5} x^{5}$

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

$32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} - x^{5}$

Expand Using Pascal's Triangle (2-x)^5

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