Finite Math Examples

${(x + 2 y)}^{2}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

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 ${(x + 2 y)}^{2}$ , $n = 2$ so the coefficients of the expansion will correspond with line $3$ .

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 $1 - 2 - 1$ .

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

Step 3

Substitute the actual values of $a$ $x$ and $b$ $2 y$ into the expression.

$1 {(x)}^{2} {(2 y)}^{0} + 2 {(x)}^{1} {(2 y)}^{1} + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(x)}^{2}$ by $1$ .

${(x)}^{2} {(2 y)}^{0} + 2 {(x)}^{1} {(2 y)}^{1} + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.2

Apply the product rule to $2 y$ .

$x^{2} (2^{0} y^{0}) + 2 {(x)}^{1} {(2 y)}^{1} + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.3

Rewrite using the commutative property of multiplication.

$2^{0} x^{2} y^{0} + 2 {(x)}^{1} {(2 y)}^{1} + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.4

Anything raised to $0$ is $1$ .

$1 x^{2} y^{0} + 2 {(x)}^{1} {(2 y)}^{1} + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.5

Multiply $x^{2}$ by $1$ .

$x^{2} y^{0} + 2 {(x)}^{1} {(2 y)}^{1} + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.6

Anything raised to $0$ is $1$ .

$x^{2} \cdot 1 + 2 {(x)}^{1} {(2 y)}^{1} + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.7

Multiply $x^{2}$ by $1$ .

$x^{2} + 2 {(x)}^{1} {(2 y)}^{1} + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.8

Simplify.

$x^{2} + 2 x {(2 y)}^{1} + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.9

Simplify.

$x^{2} + 2 x (2 y) + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.10

Rewrite using the commutative property of multiplication.

$x^{2} + 2 \cdot 2 x y + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.11

Multiply $2$ by $2$ .

$x^{2} + 4 x y + 1 {(x)}^{0} {(2 y)}^{2}$

Step 4.12

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

$x^{2} + 4 x y + {(x)}^{0} {(2 y)}^{2}$

Step 4.13

Anything raised to $0$ is $1$ .

$x^{2} + 4 x y + 1 {(2 y)}^{2}$

Step 4.14

Multiply ${(2 y)}^{2}$ by $1$ .

$x^{2} + 4 x y + {(2 y)}^{2}$

Step 4.15

Apply the product rule to $2 y$ .

$x^{2} + 4 x y + 2^{2} y^{2}$

Step 4.16

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

$x^{2} + 4 x y + 4 y^{2}$