Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Combine the opposite terms in $x \cdot x + x (- y) + y x + y (- y)$ .

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Step 2.1

Reorder the factors in the terms $x (- y)$ and $y x$ .

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

Step 2.2

Add $- x y$ and $x y$ .

$x \cdot x + 0 + y (- y) + 3 (x y - y^{2}) + 2 (x^{2} - y x)$

Step 2.3

Add $x \cdot x$ and $0$ .

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

Step 3

Simplify each term.

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Step 3.1

Multiply $x$ by $x$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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Step 3.3.1

Move $y$ .

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

Step 3.3.2

Multiply $y$ by $y$ .

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

Step 4

Apply the distributive property.

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

Step 5

Remove parentheses.

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

Step 6

Multiply $- 1$ by $3$ .

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

Step 7

Apply the distributive property.

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

Step 8

Multiply $- 1$ by $2$ .

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

Step 9

Remove parentheses.

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

Step 10

Add $x^{2}$ and $2 x^{2}$ .

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

Step 11

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

$3 x^{2} - 4 y^{2} + 3 x y - 2 y x$

Step 12

Subtract $2 y x$ from $3 x y$ .

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Step 12.1

Move $y$ .

$3 x^{2} - 4 y^{2} + 3 x y - 2 x y$

Step 12.2

Subtract $2 x y$ from $3 x y$ .

$3 x^{2} - 4 y^{2} + 1 x y$

Step 13

Rewrite $3 x^{2} - 4 y^{2} + 1 x y$ in a factored form.

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Step 13.1

Factor by grouping.

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Step 13.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

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Step 13.1.1.1

Reorder $- 4 y^{2}$ and $1 x y$ .

$3 x^{2} + 1 x y - 4 y^{2}$

Step 13.1.1.2

Factor $1$ out of $1 x y$ .

$3 x^{2} + 1 (x y) - 4 y^{2}$

Step 13.1.1.3

Rewrite $1$ as $- 3$ plus $4$

$3 x^{2} + (- 3 + 4) (x y) - 4 y^{2}$

Step 13.1.1.4

Apply the distributive property.

$3 x^{2} - 3 (x y) + 4 (x y) - 4 y^{2}$

Step 13.1.1.5

Remove unnecessary parentheses.

$3 x^{2} - 3 x y + 4 (x y) - 4 y^{2}$

Step 13.1.1.6

Remove unnecessary parentheses.

$3 x^{2} - 3 x y + 4 x y - 4 y^{2}$

Step 13.1.2

Factor out the greatest common factor from each group.

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Step 13.1.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 3 x y) + 4 x y - 4 y^{2}$

Step 13.1.2.2

Factor out the greatest common factor (GCF) from each group.

$3 x (x - 1 y) + 4 y (x - y)$

Step 13.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 1 y$ .

$(x - 1 y) (3 x + 4 y)$

Step 13.2

Rewrite $- 1 y$ as $- y$ .

$(x - y) (3 x + 4 y)$