Precalculus Examples

${(x - 2)}^{10}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

$1 - 8 - 28 - 56 - 70 - 56 - 28 - 8 - 1$

$1 - 9 - 36 - 84 - 126 - 126 - 84 - 36 - 9 - 1$

$1 - 10 - 45 - 120 - 210 - 252 - 210 - 120 - 45 - 10 - 1$

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 ${(x - 2)}^{10}$ , $n = 10$ so the coefficients of the expansion will correspond with line $11$ .

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 $1 - 10 - 45 - 120 - 210 - 252 - 210 - 120 - 45 - 10 - 1$ .

$1 a^{10} b^{0} + 10 a^{9} b + 45 a^{8} b^{2} + 120 a^{7} b^{3} + 210 a^{6} b^{4} + 252 a^{5} b^{5} + 210 a^{4} b^{6} + 120 a^{3} b^{7} + 45 a^{2} b^{8} + 10 a b^{9} + 1 a^{0} b^{10}$

Step 3

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

$1 {(x)}^{10} {(- 2)}^{0} + 10 {(x)}^{9} {(- 2)}^{1} + 45 {(x)}^{8} {(- 2)}^{2} + 120 {(x)}^{7} {(- 2)}^{3} + 210 {(x)}^{6} {(- 2)}^{4} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(x)}^{10}$ by $1$ .

${(x)}^{10} {(- 2)}^{0} + 10 {(x)}^{9} {(- 2)}^{1} + 45 {(x)}^{8} {(- 2)}^{2} + 120 {(x)}^{7} {(- 2)}^{3} + 210 {(x)}^{6} {(- 2)}^{4} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.2

Anything raised to $0$ is $1$ .

$x^{10} \cdot 1 + 10 {(x)}^{9} {(- 2)}^{1} + 45 {(x)}^{8} {(- 2)}^{2} + 120 {(x)}^{7} {(- 2)}^{3} + 210 {(x)}^{6} {(- 2)}^{4} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.3

Multiply $x^{10}$ by $1$ .

$x^{10} + 10 {(x)}^{9} {(- 2)}^{1} + 45 {(x)}^{8} {(- 2)}^{2} + 120 {(x)}^{7} {(- 2)}^{3} + 210 {(x)}^{6} {(- 2)}^{4} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.4

Evaluate the exponent.

$x^{10} + 10 x^{9} \cdot - 2 + 45 {(x)}^{8} {(- 2)}^{2} + 120 {(x)}^{7} {(- 2)}^{3} + 210 {(x)}^{6} {(- 2)}^{4} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.5

Multiply $- 2$ by $10$ .

$x^{10} - 20 x^{9} + 45 {(x)}^{8} {(- 2)}^{2} + 120 {(x)}^{7} {(- 2)}^{3} + 210 {(x)}^{6} {(- 2)}^{4} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.6

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

$x^{10} - 20 x^{9} + 45 x^{8} \cdot 4 + 120 {(x)}^{7} {(- 2)}^{3} + 210 {(x)}^{6} {(- 2)}^{4} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.7

Multiply $4$ by $45$ .

$x^{10} - 20 x^{9} + 180 x^{8} + 120 {(x)}^{7} {(- 2)}^{3} + 210 {(x)}^{6} {(- 2)}^{4} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.8

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

$x^{10} - 20 x^{9} + 180 x^{8} + 120 x^{7} \cdot - 8 + 210 {(x)}^{6} {(- 2)}^{4} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.9

Multiply $- 8$ by $120$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 210 {(x)}^{6} {(- 2)}^{4} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.10

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

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 210 x^{6} \cdot 16 + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.11

Multiply $16$ by $210$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} + 252 {(x)}^{5} {(- 2)}^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.12

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

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} + 252 x^{5} \cdot - 32 + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.13

Multiply $- 32$ by $252$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 210 {(x)}^{4} {(- 2)}^{6} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.14

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

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 210 x^{4} \cdot 64 + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.15

Multiply $64$ by $210$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} + 120 {(x)}^{3} {(- 2)}^{7} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.16

Raise $- 2$ to the power of $7$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} + 120 x^{3} \cdot - 128 + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.17

Multiply $- 128$ by $120$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} - 15360 x^{3} + 45 {(x)}^{2} {(- 2)}^{8} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.18

Raise $- 2$ to the power of $8$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} - 15360 x^{3} + 45 x^{2} \cdot 256 + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.19

Multiply $256$ by $45$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} - 15360 x^{3} + 11520 x^{2} + 10 {(x)}^{1} {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.20

Simplify.

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} - 15360 x^{3} + 11520 x^{2} + 10 x {(- 2)}^{9} + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.21

Raise $- 2$ to the power of $9$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} - 15360 x^{3} + 11520 x^{2} + 10 x \cdot - 512 + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.22

Multiply $- 512$ by $10$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} - 15360 x^{3} + 11520 x^{2} - 5120 x + 1 {(x)}^{0} {(- 2)}^{10}$

Step 4.23

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

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} - 15360 x^{3} + 11520 x^{2} - 5120 x + {(x)}^{0} {(- 2)}^{10}$

Step 4.24

Anything raised to $0$ is $1$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} - 15360 x^{3} + 11520 x^{2} - 5120 x + 1 {(- 2)}^{10}$

Step 4.25

Multiply ${(- 2)}^{10}$ by $1$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} - 15360 x^{3} + 11520 x^{2} - 5120 x + {(- 2)}^{10}$

Step 4.26

Raise $- 2$ to the power of $10$ .

$x^{10} - 20 x^{9} + 180 x^{8} - 960 x^{7} + 3360 x^{6} - 8064 x^{5} + 13440 x^{4} - 15360 x^{3} + 11520 x^{2} - 5120 x + 1024$