Expand Using Pascal's Triangle (x-3)^6

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

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 ${(x - 3)}^{6}$ , $n = 6$ 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 $a$ $x$ and $b$ $- 3$ into the expression.

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

Step 4

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

Multiply $x^{6}$ by $1$ .

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

Evaluate the exponent.

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

Multiply $- 3$ by $6$ .

$x^{6} - 18 x^{5} + 15 {(x)}^{4} {(- 3)}^{2} + 20 {(x)}^{3} {(- 3)}^{3} + 15 {(x)}^{2} {(- 3)}^{4} + 6 {(x)}^{1} {(- 3)}^{5} + 1 {(x)}^{0} {(- 3)}^{6}$

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

$x^{6} - 18 x^{5} + 15 x^{4} \cdot 9 + 20 {(x)}^{3} {(- 3)}^{3} + 15 {(x)}^{2} {(- 3)}^{4} + 6 {(x)}^{1} {(- 3)}^{5} + 1 {(x)}^{0} {(- 3)}^{6}$

Multiply $9$ by $15$ .

$x^{6} - 18 x^{5} + 135 x^{4} + 20 {(x)}^{3} {(- 3)}^{3} + 15 {(x)}^{2} {(- 3)}^{4} + 6 {(x)}^{1} {(- 3)}^{5} + 1 {(x)}^{0} {(- 3)}^{6}$

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

$x^{6} - 18 x^{5} + 135 x^{4} + 20 x^{3} \cdot - 27 + 15 {(x)}^{2} {(- 3)}^{4} + 6 {(x)}^{1} {(- 3)}^{5} + 1 {(x)}^{0} {(- 3)}^{6}$

Multiply $- 27$ by $20$ .

$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 15 {(x)}^{2} {(- 3)}^{4} + 6 {(x)}^{1} {(- 3)}^{5} + 1 {(x)}^{0} {(- 3)}^{6}$

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

$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 15 x^{2} \cdot 81 + 6 {(x)}^{1} {(- 3)}^{5} + 1 {(x)}^{0} {(- 3)}^{6}$

Multiply $81$ by $15$ .

$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 1215 x^{2} + 6 {(x)}^{1} {(- 3)}^{5} + 1 {(x)}^{0} {(- 3)}^{6}$

Simplify.

$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 1215 x^{2} + 6 x {(- 3)}^{5} + 1 {(x)}^{0} {(- 3)}^{6}$

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

$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 1215 x^{2} + 6 x \cdot - 243 + 1 {(x)}^{0} {(- 3)}^{6}$

Multiply $- 243$ by $6$ .

$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 1215 x^{2} - 1458 x + 1 {(x)}^{0} {(- 3)}^{6}$

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

$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 1215 x^{2} - 1458 x + {(x)}^{0} {(- 3)}^{6}$

Anything raised to $0$ is $1$ .

$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 1215 x^{2} - 1458 x + 1 {(- 3)}^{6}$

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

$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 1215 x^{2} - 1458 x + {(- 3)}^{6}$

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

$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 1215 x^{2} - 1458 x + 729$

Expand Using Pascal's Triangle (x-3)^6

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