Finite Math Examples

$(x - 2 y) (x - y) (x + 3 y)$

Step 1

Expand $(x - 2 y) (x - y)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$(x (x - y) - 2 y (x - y)) (x + 3 y)$

Step 1.2

Apply the distributive property.

$(x \cdot x + x (- y) - 2 y (x - y)) (x + 3 y)$

Step 1.3

Apply the distributive property.

$(x \cdot x + x (- y) - 2 y x - 2 y (- y)) (x + 3 y)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $x$ by $x$ .

$(x^{2} + x (- y) - 2 y x - 2 y (- y)) (x + 3 y)$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$(x^{2} - x y - 2 y x - 2 y (- y)) (x + 3 y)$

Step 2.1.3

Rewrite using the commutative property of multiplication.

$(x^{2} - x y - 2 y x - 2 \cdot - 1 y \cdot y) (x + 3 y)$

Step 2.1.4

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.4.1

Move $y$ .

$(x^{2} - x y - 2 y x - 2 \cdot - 1 (y \cdot y)) (x + 3 y)$

Step 2.1.4.2

Multiply $y$ by $y$ .

$(x^{2} - x y - 2 y x - 2 \cdot - 1 y^{2}) (x + 3 y)$

Step 2.1.5

Multiply $- 2$ by $- 1$ .

$(x^{2} - x y - 2 y x + 2 y^{2}) (x + 3 y)$

Step 2.2

Subtract $2 y x$ from $- x y$ .

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Step 2.2.1

Move $y$ .

$(x^{2} - x y - 2 x y + 2 y^{2}) (x + 3 y)$

Step 2.2.2

Subtract $2 x y$ from $- x y$ .

$(x^{2} - 3 x y + 2 y^{2}) (x + 3 y)$

Step 3

Expand $(x^{2} - 3 x y + 2 y^{2}) (x + 3 y)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x + x^{2} (3 y) - 3 x y x - 3 x y (3 y) + 2 y^{2} x + 2 y^{2} (3 y)$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.1.1.1

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

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Step 4.1.1.1.1

Raise $x$ to the power of $1$ .

$x^{2} x^{1} + x^{2} (3 y) - 3 x y x - 3 x y (3 y) + 2 y^{2} x + 2 y^{2} (3 y)$

Step 4.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 1} + x^{2} (3 y) - 3 x y x - 3 x y (3 y) + 2 y^{2} x + 2 y^{2} (3 y)$

Step 4.1.1.2

Add $2$ and $1$ .

$x^{3} + x^{2} (3 y) - 3 x y x - 3 x y (3 y) + 2 y^{2} x + 2 y^{2} (3 y)$

Step 4.1.2

Rewrite using the commutative property of multiplication.

$x^{3} + 3 x^{2} y - 3 x y x - 3 x y (3 y) + 2 y^{2} x + 2 y^{2} (3 y)$

Step 4.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.3.1

Move $x$ .

$x^{3} + 3 x^{2} y - 3 (x \cdot x) y - 3 x y (3 y) + 2 y^{2} x + 2 y^{2} (3 y)$

Step 4.1.3.2

Multiply $x$ by $x$ .

$x^{3} + 3 x^{2} y - 3 x^{2} y - 3 x y (3 y) + 2 y^{2} x + 2 y^{2} (3 y)$

Step 4.1.4

Multiply $y$ by $y$ by adding the exponents.

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Step 4.1.4.1

Move $y$ .

$x^{3} + 3 x^{2} y - 3 x^{2} y - 3 x (y \cdot y) \cdot 3 + 2 y^{2} x + 2 y^{2} (3 y)$

Step 4.1.4.2

Multiply $y$ by $y$ .

$x^{3} + 3 x^{2} y - 3 x^{2} y - 3 x y^{2} \cdot 3 + 2 y^{2} x + 2 y^{2} (3 y)$

Step 4.1.5

Multiply $3$ by $- 3$ .

$x^{3} + 3 x^{2} y - 3 x^{2} y - 9 x y^{2} + 2 y^{2} x + 2 y^{2} (3 y)$

Step 4.1.6

Rewrite using the commutative property of multiplication.

$x^{3} + 3 x^{2} y - 3 x^{2} y - 9 x y^{2} + 2 y^{2} x + 2 \cdot 3 y^{2} y$

Step 4.1.7

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 4.1.7.1

Move $y$ .

$x^{3} + 3 x^{2} y - 3 x^{2} y - 9 x y^{2} + 2 y^{2} x + 2 \cdot 3 (y \cdot y^{2})$

Step 4.1.7.2

Multiply $y$ by $y^{2}$ .

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Step 4.1.7.2.1

Raise $y$ to the power of $1$ .

$x^{3} + 3 x^{2} y - 3 x^{2} y - 9 x y^{2} + 2 y^{2} x + 2 \cdot 3 (y^{1} y^{2})$

Step 4.1.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} + 3 x^{2} y - 3 x^{2} y - 9 x y^{2} + 2 y^{2} x + 2 \cdot 3 y^{1 + 2}$

Step 4.1.7.3

Add $1$ and $2$ .

$x^{3} + 3 x^{2} y - 3 x^{2} y - 9 x y^{2} + 2 y^{2} x + 2 \cdot 3 y^{3}$

Step 4.1.8

Multiply $2$ by $3$ .

$x^{3} + 3 x^{2} y - 3 x^{2} y - 9 x y^{2} + 2 y^{2} x + 6 y^{3}$

Step 4.2

Combine the opposite terms in $x^{3} + 3 x^{2} y - 3 x^{2} y - 9 x y^{2} + 2 y^{2} x + 6 y^{3}$ .

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Step 4.2.1

Subtract $3 x^{2} y$ from $3 x^{2} y$ .

$x^{3} + 0 - 9 x y^{2} + 2 y^{2} x + 6 y^{3}$

Step 4.2.2

Add $x^{3}$ and $0$ .

$x^{3} - 9 x y^{2} + 2 y^{2} x + 6 y^{3}$

Step 5

Add $- 9 x y^{2}$ and $2 y^{2} x$ .

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Step 5.1

Move $y^{2}$ .

$x^{3} - 9 x y^{2} + 2 x y^{2} + 6 y^{3}$

Step 5.2

Add $- 9 x y^{2}$ and $2 x y^{2}$ .

$x^{3} - 7 x y^{2} + 6 y^{3}$