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

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

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 ${(x - 2)}^{5}$ , $n = 5$ 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 $a$ $x$ and $b$ $- 2$ into the expression.

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

Step 4

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

Multiply $x^{5}$ by $1$ .

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

Evaluate the exponent.

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

Multiply $- 2$ by $5$ .

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

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

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

Multiply $4$ by $10$ .

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

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

$x^{5} - 10 x^{4} + 40 x^{3} + 10 x^{2} \cdot - 8 + 5 {(x)}^{1} {(- 2)}^{4} + 1 {(x)}^{0} {(- 2)}^{5}$

Multiply $- 8$ by $10$ .

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

Simplify.

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

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

$x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 5 x \cdot 16 + 1 {(x)}^{0} {(- 2)}^{5}$

Multiply $16$ by $5$ .

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

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

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

Anything raised to $0$ is $1$ .

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

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

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

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

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

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

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