Algebra Examples

${(3 x - 2)}^{10}$

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 ${(3 x - 2)}^{10}$ , $n = 10$ 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 $a$ $3 x$ and $b$ $- 2$ into the expression.

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

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply ${(3 x)}^{10}$ by $1$ .

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

Step 4.2

Apply the product rule to $3 x$ .

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

Step 4.3

Raise $3$ to the power of $10$ .

$59049 x^{10} {(- 2)}^{0} + 10 {(3 x)}^{9} {(- 2)}^{1} + 45 {(3 x)}^{8} {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.4

Anything raised to $0$ is $1$ .

$59049 x^{10} \cdot 1 + 10 {(3 x)}^{9} {(- 2)}^{1} + 45 {(3 x)}^{8} {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.5

Multiply $59049$ by $1$ .

$59049 x^{10} + 10 {(3 x)}^{9} {(- 2)}^{1} + 45 {(3 x)}^{8} {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.6

Apply the product rule to $3 x$ .

$59049 x^{10} + 10 (3^{9} x^{9}) {(- 2)}^{1} + 45 {(3 x)}^{8} {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.7

Raise $3$ to the power of $9$ .

$59049 x^{10} + 10 (19683 x^{9}) {(- 2)}^{1} + 45 {(3 x)}^{8} {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.8

Multiply $19683$ by $10$ .

$59049 x^{10} + 196830 x^{9} {(- 2)}^{1} + 45 {(3 x)}^{8} {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.9

Evaluate the exponent.

$59049 x^{10} + 196830 x^{9} \cdot - 2 + 45 {(3 x)}^{8} {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.10

Multiply $- 2$ by $196830$ .

$59049 x^{10} - 393660 x^{9} + 45 {(3 x)}^{8} {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.11

Apply the product rule to $3 x$ .

$59049 x^{10} - 393660 x^{9} + 45 (3^{8} x^{8}) {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.12

Raise $3$ to the power of $8$ .

$59049 x^{10} - 393660 x^{9} + 45 (6561 x^{8}) {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.13

Multiply $6561$ by $45$ .

$59049 x^{10} - 393660 x^{9} + 295245 x^{8} {(- 2)}^{2} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.14

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

$59049 x^{10} - 393660 x^{9} + 295245 x^{8} \cdot 4 + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.15

Multiply $4$ by $295245$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} + 120 {(3 x)}^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.16

Apply the product rule to $3 x$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} + 120 (3^{7} x^{7}) {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.17

Raise $3$ to the power of $7$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} + 120 (2187 x^{7}) {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.18

Multiply $2187$ by $120$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} + 262440 x^{7} {(- 2)}^{3} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.19

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} + 262440 x^{7} \cdot - 8 + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.20

Multiply $- 8$ by $262440$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 210 {(3 x)}^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.21

Apply the product rule to $3 x$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 210 (3^{6} x^{6}) {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.22

Raise $3$ to the power of $6$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 210 (729 x^{6}) {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.23

Multiply $729$ by $210$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 153090 x^{6} {(- 2)}^{4} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.24

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 153090 x^{6} \cdot 16 + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.25

Multiply $16$ by $153090$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} + 252 {(3 x)}^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.26

Apply the product rule to $3 x$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} + 252 (3^{5} x^{5}) {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.27

Raise $3$ to the power of $5$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} + 252 (243 x^{5}) {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.28

Multiply $243$ by $252$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} + 61236 x^{5} {(- 2)}^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.29

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} + 61236 x^{5} \cdot - 32 + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.30

Multiply $- 32$ by $61236$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 210 {(3 x)}^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.31

Apply the product rule to $3 x$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 210 (3^{4} x^{4}) {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.32

Raise $3$ to the power of $4$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 210 (81 x^{4}) {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.33

Multiply $81$ by $210$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 17010 x^{4} {(- 2)}^{6} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.34

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 17010 x^{4} \cdot 64 + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.35

Multiply $64$ by $17010$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} + 120 {(3 x)}^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.36

Apply the product rule to $3 x$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} + 120 (3^{3} x^{3}) {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.37

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} + 120 (27 x^{3}) {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.38

Multiply $27$ by $120$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} + 3240 x^{3} {(- 2)}^{7} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.39

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} + 3240 x^{3} \cdot - 128 + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.40

Multiply $- 128$ by $3240$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 45 {(3 x)}^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.41

Apply the product rule to $3 x$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 45 (3^{2} x^{2}) {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.42

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 45 (9 x^{2}) {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.43

Multiply $9$ by $45$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 405 x^{2} {(- 2)}^{8} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.44

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 405 x^{2} \cdot 256 + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.45

Multiply $256$ by $405$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} + 10 {(3 x)}^{1} {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.46

Simplify.

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} + 10 (3 x) {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.47

Multiply $3$ by $10$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} + 30 x {(- 2)}^{9} + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.48

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} + 30 x \cdot - 512 + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.49

Multiply $- 512$ by $30$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} - 15360 x + 1 {(3 x)}^{0} {(- 2)}^{10}$

Step 4.50

Multiply ${(3 x)}^{0}$ by $1$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} - 15360 x + {(3 x)}^{0} {(- 2)}^{10}$

Step 4.51

Apply the product rule to $3 x$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} - 15360 x + 3^{0} x^{0} {(- 2)}^{10}$

Step 4.52

Anything raised to $0$ is $1$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} - 15360 x + 1 x^{0} {(- 2)}^{10}$

Step 4.53

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} - 15360 x + x^{0} {(- 2)}^{10}$

Step 4.54

Anything raised to $0$ is $1$ .

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} - 15360 x + 1 {(- 2)}^{10}$

Step 4.55

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} - 15360 x + {(- 2)}^{10}$

Step 4.56

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

$59049 x^{10} - 393660 x^{9} + 1180980 x^{8} - 2099520 x^{7} + 2449440 x^{6} - 1959552 x^{5} + 1088640 x^{4} - 414720 x^{3} + 103680 x^{2} - 15360 x + 1024$