Precalculus Examples

${(q + r)}^{3}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

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 ${(q + r)}^{3}$ , $n = 3$ so the coefficients of the expansion will correspond with line $4$ .

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 $1 - 3 - 3 - 1$ .

$1 a^{3} b^{0} + 3 a^{2} b + 3 a b^{2} + 1 a^{0} b^{3}$

Step 3

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

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Anything raised to $0$ is $1$ .

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

Step 4.3

Multiply $q^{3}$ by $1$ .

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

Step 4.4

Simplify.

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

Step 4.5

Simplify.

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

Step 4.6

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

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

Step 4.7

Anything raised to $0$ is $1$ .

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

Step 4.8

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

$q^{3} + 3 q^{2} r + 3 q r^{2} + r^{3}$