Expand Using Pascal's Triangle (z-4)^7

${(z - 4)}^{7}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

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

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

$1 - 7 - 21 - 35 - 35 - 21 - 7 - 1$

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 ${(z - 4)}^{7}$ , $n = 7$ so the coefficients of the expansion will correspond with line $8$ .

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 $1 - 7 - 21 - 35 - 35 - 21 - 7 - 1$ .

$1 a^{7} b^{0} + 7 a^{6} b + 21 a^{5} b^{2} + 35 a^{4} b^{3} + 35 a^{3} b^{4} + 21 a^{2} b^{5} + 7 a b^{6} + 1 a^{0} b^{7}$

Step 3

Substitute the actual values of $a$ $z$ and $b$ $- 4$ into the expression.

$1 {(z)}^{7} {(- 4)}^{0} + 7 {(z)}^{6} {(- 4)}^{1} + 21 {(z)}^{5} {(- 4)}^{2} + 35 {(z)}^{4} {(- 4)}^{3} + 35 {(z)}^{3} {(- 4)}^{4} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Step 4

Simplify each term.

Multiply ${(z)}^{7}$ by $1$ .

${(z)}^{7} {(- 4)}^{0} + 7 {(z)}^{6} {(- 4)}^{1} + 21 {(z)}^{5} {(- 4)}^{2} + 35 {(z)}^{4} {(- 4)}^{3} + 35 {(z)}^{3} {(- 4)}^{4} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Anything raised to $0$ is $1$ .

$z^{7} \cdot 1 + 7 {(z)}^{6} {(- 4)}^{1} + 21 {(z)}^{5} {(- 4)}^{2} + 35 {(z)}^{4} {(- 4)}^{3} + 35 {(z)}^{3} {(- 4)}^{4} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Multiply $z^{7}$ by $1$ .

$z^{7} + 7 {(z)}^{6} {(- 4)}^{1} + 21 {(z)}^{5} {(- 4)}^{2} + 35 {(z)}^{4} {(- 4)}^{3} + 35 {(z)}^{3} {(- 4)}^{4} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Evaluate the exponent.

$z^{7} + 7 z^{6} \cdot - 4 + 21 {(z)}^{5} {(- 4)}^{2} + 35 {(z)}^{4} {(- 4)}^{3} + 35 {(z)}^{3} {(- 4)}^{4} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Multiply $- 4$ by $7$ .

$z^{7} - 28 z^{6} + 21 {(z)}^{5} {(- 4)}^{2} + 35 {(z)}^{4} {(- 4)}^{3} + 35 {(z)}^{3} {(- 4)}^{4} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

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

$z^{7} - 28 z^{6} + 21 z^{5} \cdot 16 + 35 {(z)}^{4} {(- 4)}^{3} + 35 {(z)}^{3} {(- 4)}^{4} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Multiply $16$ by $21$ .

$z^{7} - 28 z^{6} + 336 z^{5} + 35 {(z)}^{4} {(- 4)}^{3} + 35 {(z)}^{3} {(- 4)}^{4} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

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

$z^{7} - 28 z^{6} + 336 z^{5} + 35 z^{4} \cdot - 64 + 35 {(z)}^{3} {(- 4)}^{4} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Multiply $- 64$ by $35$ .

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 35 {(z)}^{3} {(- 4)}^{4} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

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

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 35 z^{3} \cdot 256 + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Multiply $256$ by $35$ .

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 8960 z^{3} + 21 {(z)}^{2} {(- 4)}^{5} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Raise $- 4$ to the power of $5$ .

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 8960 z^{3} + 21 z^{2} \cdot - 1024 + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Multiply $- 1024$ by $21$ .

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 8960 z^{3} - 21504 z^{2} + 7 {(z)}^{1} {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Simplify.

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 8960 z^{3} - 21504 z^{2} + 7 z {(- 4)}^{6} + 1 {(z)}^{0} {(- 4)}^{7}$

Raise $- 4$ to the power of $6$ .

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 8960 z^{3} - 21504 z^{2} + 7 z \cdot 4096 + 1 {(z)}^{0} {(- 4)}^{7}$

Multiply $4096$ by $7$ .

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 8960 z^{3} - 21504 z^{2} + 28672 z + 1 {(z)}^{0} {(- 4)}^{7}$

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

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 8960 z^{3} - 21504 z^{2} + 28672 z + {(z)}^{0} {(- 4)}^{7}$

Anything raised to $0$ is $1$ .

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 8960 z^{3} - 21504 z^{2} + 28672 z + 1 {(- 4)}^{7}$

Multiply ${(- 4)}^{7}$ by $1$ .

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 8960 z^{3} - 21504 z^{2} + 28672 z + {(- 4)}^{7}$

Raise $- 4$ to the power of $7$ .

$z^{7} - 28 z^{6} + 336 z^{5} - 2240 z^{4} + 8960 z^{3} - 21504 z^{2} + 28672 z - 16384$

Expand Using Pascal's Triangle (z-4)^7

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