Algebra Examples

${(3 - 2 x)}^{6}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

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

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 ${(3 - 2 x)}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

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

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

Step 3

Substitute the actual values of $a$ $3$ and $b$ $- 2 x$ into the expression.

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

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply ${(3)}^{6}$ by $1$ .

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

Step 4.2

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

$729 {(- 2 x)}^{0} + 6 {(3)}^{5} {(- 2 x)}^{1} + 15 {(3)}^{4} {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.3

Apply the product rule to $- 2 x$ .

$729 ({(- 2)}^{0} x^{0}) + 6 {(3)}^{5} {(- 2 x)}^{1} + 15 {(3)}^{4} {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.4

Anything raised to $0$ is $1$ .

$729 (1 x^{0}) + 6 {(3)}^{5} {(- 2 x)}^{1} + 15 {(3)}^{4} {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.5

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

$729 x^{0} + 6 {(3)}^{5} {(- 2 x)}^{1} + 15 {(3)}^{4} {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.6

Anything raised to $0$ is $1$ .

$729 \cdot 1 + 6 {(3)}^{5} {(- 2 x)}^{1} + 15 {(3)}^{4} {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.7

Multiply $729$ by $1$ .

$729 + 6 {(3)}^{5} {(- 2 x)}^{1} + 15 {(3)}^{4} {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.8

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

$729 + 6 \cdot 243 {(- 2 x)}^{1} + 15 {(3)}^{4} {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.9

Multiply $6$ by $243$ .

$729 + 1458 {(- 2 x)}^{1} + 15 {(3)}^{4} {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.10

Simplify.

$729 + 1458 (- 2 x) + 15 {(3)}^{4} {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.11

Multiply $- 2$ by $1458$ .

$729 - 2916 x + 15 {(3)}^{4} {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.12

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

$729 - 2916 x + 15 \cdot 81 {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.13

Multiply $15$ by $81$ .

$729 - 2916 x + 1215 {(- 2 x)}^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.14

Apply the product rule to $- 2 x$ .

$729 - 2916 x + 1215 ({(- 2)}^{2} x^{2}) + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.15

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

$729 - 2916 x + 1215 (4 x^{2}) + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.16

Multiply $4$ by $1215$ .

$729 - 2916 x + 4860 x^{2} + 20 {(3)}^{3} {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.17

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

$729 - 2916 x + 4860 x^{2} + 20 \cdot 27 {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.18

Multiply $20$ by $27$ .

$729 - 2916 x + 4860 x^{2} + 540 {(- 2 x)}^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.19

Apply the product rule to $- 2 x$ .

$729 - 2916 x + 4860 x^{2} + 540 ({(- 2)}^{3} x^{3}) + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.20

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

$729 - 2916 x + 4860 x^{2} + 540 (- 8 x^{3}) + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.21

Multiply $- 8$ by $540$ .

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 15 {(3)}^{2} {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.22

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

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 15 \cdot 9 {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.23

Multiply $15$ by $9$ .

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 135 {(- 2 x)}^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.24

Apply the product rule to $- 2 x$ .

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 135 ({(- 2)}^{4} x^{4}) + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.25

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

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 135 (16 x^{4}) + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.26

Multiply $16$ by $135$ .

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} + 6 {(3)}^{1} {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.27

Evaluate the exponent.

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} + 6 \cdot 3 {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.28

Multiply $6$ by $3$ .

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} + 18 {(- 2 x)}^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.29

Apply the product rule to $- 2 x$ .

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} + 18 ({(- 2)}^{5} x^{5}) + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.30

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

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} + 18 (- 32 x^{5}) + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.31

Multiply $- 32$ by $18$ .

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} - 576 x^{5} + 1 {(3)}^{0} {(- 2 x)}^{6}$

Step 4.32

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

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} - 576 x^{5} + {(3)}^{0} {(- 2 x)}^{6}$

Step 4.33

Anything raised to $0$ is $1$ .

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} - 576 x^{5} + 1 {(- 2 x)}^{6}$

Step 4.34

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

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} - 576 x^{5} + {(- 2 x)}^{6}$

Step 4.35

Apply the product rule to $- 2 x$ .

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} - 576 x^{5} + {(- 2)}^{6} x^{6}$

Step 4.36

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

$729 - 2916 x + 4860 x^{2} - 4320 x^{3} + 2160 x^{4} - 576 x^{5} + 64 x^{6}$