Expand Using Pascal's Triangle (q+r)^5

${(q + r)}^{5}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

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 ${(q + r)}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

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

$1 a^{5} b^{0} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 a^{0} b^{5}$

Step 3

Substitute the actual values of $a$ $q$ and $b$ $r$ into the expression.

$1 {(q)}^{5} {(r)}^{0} + 5 {(q)}^{4} {(r)}^{1} + 10 {(q)}^{3} {(r)}^{2} + 10 {(q)}^{2} {(r)}^{3} + 5 {(q)}^{1} {(r)}^{4} + 1 {(q)}^{0} {(r)}^{5}$

Step 4

Simplify each term.

Multiply ${(q)}^{5}$ by $1$ .

${(q)}^{5} {(r)}^{0} + 5 {(q)}^{4} {(r)}^{1} + 10 {(q)}^{3} {(r)}^{2} + 10 {(q)}^{2} {(r)}^{3} + 5 {(q)}^{1} {(r)}^{4} + 1 {(q)}^{0} {(r)}^{5}$

Anything raised to $0$ is $1$ .

$q^{5} \cdot 1 + 5 {(q)}^{4} {(r)}^{1} + 10 {(q)}^{3} {(r)}^{2} + 10 {(q)}^{2} {(r)}^{3} + 5 {(q)}^{1} {(r)}^{4} + 1 {(q)}^{0} {(r)}^{5}$

Multiply $q^{5}$ by $1$ .

$q^{5} + 5 {(q)}^{4} {(r)}^{1} + 10 {(q)}^{3} {(r)}^{2} + 10 {(q)}^{2} {(r)}^{3} + 5 {(q)}^{1} {(r)}^{4} + 1 {(q)}^{0} {(r)}^{5}$

Simplify.

$q^{5} + 5 q^{4} r + 10 {(q)}^{3} {(r)}^{2} + 10 {(q)}^{2} {(r)}^{3} + 5 {(q)}^{1} {(r)}^{4} + 1 {(q)}^{0} {(r)}^{5}$

Simplify.

$q^{5} + 5 q^{4} r + 10 q^{3} r^{2} + 10 q^{2} r^{3} + 5 q {(r)}^{4} + 1 {(q)}^{0} {(r)}^{5}$

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

$q^{5} + 5 q^{4} r + 10 q^{3} r^{2} + 10 q^{2} r^{3} + 5 q r^{4} + {(q)}^{0} {(r)}^{5}$

Anything raised to $0$ is $1$ .

$q^{5} + 5 q^{4} r + 10 q^{3} r^{2} + 10 q^{2} r^{3} + 5 q r^{4} + 1 {(r)}^{5}$

Multiply ${(r)}^{5}$ by $1$ .

$q^{5} + 5 q^{4} r + 10 q^{3} r^{2} + 10 q^{2} r^{3} + 5 q r^{4} + r^{5}$

Expand Using Pascal's Triangle (q+r)^5

Related Questions

Support LCMGCF.com by disabling your adblocker.