Expand Using Pascal's Triangle (d-5)^6

${(d - 5)}^{6}$

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 ${(d - 5)}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

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

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

Substitute the actual values of $a$ $d$ and $b$ $- 5$ into the expression.

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

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

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

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

Evaluate the exponent.

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

Multiply $- 5$ by $6$ .

$d^{6} - 30 d^{5} + 15 {(d)}^{4} {(- 5)}^{2} + 20 {(d)}^{3} {(- 5)}^{3} + 15 {(d)}^{2} {(- 5)}^{4} + 6 {(d)}^{1} {(- 5)}^{5} + 1 {(d)}^{0} {(- 5)}^{6}$

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

$d^{6} - 30 d^{5} + 15 d^{4} \cdot 25 + 20 {(d)}^{3} {(- 5)}^{3} + 15 {(d)}^{2} {(- 5)}^{4} + 6 {(d)}^{1} {(- 5)}^{5} + 1 {(d)}^{0} {(- 5)}^{6}$

Multiply $25$ by $15$ .

$d^{6} - 30 d^{5} + 375 d^{4} + 20 {(d)}^{3} {(- 5)}^{3} + 15 {(d)}^{2} {(- 5)}^{4} + 6 {(d)}^{1} {(- 5)}^{5} + 1 {(d)}^{0} {(- 5)}^{6}$

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

$d^{6} - 30 d^{5} + 375 d^{4} + 20 d^{3} \cdot - 125 + 15 {(d)}^{2} {(- 5)}^{4} + 6 {(d)}^{1} {(- 5)}^{5} + 1 {(d)}^{0} {(- 5)}^{6}$

Multiply $- 125$ by $20$ .

$d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 15 {(d)}^{2} {(- 5)}^{4} + 6 {(d)}^{1} {(- 5)}^{5} + 1 {(d)}^{0} {(- 5)}^{6}$

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

$d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 15 d^{2} \cdot 625 + 6 {(d)}^{1} {(- 5)}^{5} + 1 {(d)}^{0} {(- 5)}^{6}$

Multiply $625$ by $15$ .

$d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 9375 d^{2} + 6 {(d)}^{1} {(- 5)}^{5} + 1 {(d)}^{0} {(- 5)}^{6}$

Simplify.

$d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 9375 d^{2} + 6 d {(- 5)}^{5} + 1 {(d)}^{0} {(- 5)}^{6}$

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

$d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 9375 d^{2} + 6 d \cdot - 3125 + 1 {(d)}^{0} {(- 5)}^{6}$

Multiply $- 3125$ by $6$ .

$d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 9375 d^{2} - 18750 d + 1 {(d)}^{0} {(- 5)}^{6}$

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

$d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 9375 d^{2} - 18750 d + {(d)}^{0} {(- 5)}^{6}$

Anything raised to $0$ is $1$ .

$d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 9375 d^{2} - 18750 d + 1 {(- 5)}^{6}$

Multiply ${(- 5)}^{6}$ by $1$ .

$d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 9375 d^{2} - 18750 d + {(- 5)}^{6}$

Raise $- 5$ to the power of $6$ .

$d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 9375 d^{2} - 18750 d + 15625$

Expand Using Pascal's Triangle (d-5)^6

Related Questions

Support LCMGCF.com by disabling your adblocker.