Algebra Examples

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

Step 1

Factor $y$ out of $x y (x - y) + 3 y {(y - x)}^{2}$ .

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Step 1.1

Factor $y$ out of $x y (x - y)$ .

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

Step 1.2

Factor $y$ out of $3 y {(y - x)}^{2}$ .

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

Step 1.3

Factor $y$ out of $y (x (x - y)) + y (3 {(y - x)}^{2})$ .

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

Step 2

Apply the distributive property.

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

Step 3

Multiply $x$ by $x$ .

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

Step 4

Rewrite using the commutative property of multiplication.

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

Step 5

Rewrite ${(y - x)}^{2}$ as $(y - x) (y - x)$ .

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

Step 6

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

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $y$ by $y$ .

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

Step 7.1.2

Rewrite using the commutative property of multiplication.

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

Step 7.1.3

Rewrite using the commutative property of multiplication.

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

Step 7.1.4

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

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Step 7.1.4.1

Move $x$ .

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

Step 7.1.4.2

Multiply $x$ by $x$ .

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

Step 7.1.5

Multiply $- 1$ by $- 1$ .

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

Step 7.1.6

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

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

Step 7.2

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

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Step 7.2.1

Move $y$ .

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

Step 7.2.2

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

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

Step 8

Apply the distributive property.

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

Step 9

Multiply $- 2$ by $3$ .

$y (x^{2} - x y + 3 y^{2} - 6 (x y) + 3 x^{2})$

Step 10

Remove parentheses.

$y (x^{2} - x y + 3 y^{2} - 6 x y + 3 x^{2})$

Step 11

Add $x^{2}$ and $3 x^{2}$ .

$y (4 x^{2} - x y + 3 y^{2} - 6 x y)$

Step 12

Subtract $6 x y$ from $- x y$ .

$y (4 x^{2} + 3 y^{2} - 7 x y)$

Step 13

Factor.

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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 $a \cdot c = 4 \cdot 3 = 12$ and whose sum is $b = - 7$ .

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Step 13.1.1.1

Reorder $3 y^{2}$ and $- 7 x y$ .

$y (4 x^{2} - 7 x y + 3 y^{2})$

Step 13.1.1.2

Factor $- 7$ out of $- 7 x y$ .

$y (4 x^{2} - 7 (x y) + 3 y^{2})$

Step 13.1.1.3

Rewrite $- 7$ as $- 3$ plus $- 4$

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

Step 13.1.1.4

Apply the distributive property.

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

Step 13.1.1.5

Remove unnecessary parentheses.

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

Step 13.1.1.6

Remove unnecessary parentheses.

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

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.

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

Step 13.1.2.2

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

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

Step 13.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 3 y$ .

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

Step 13.2

Remove unnecessary parentheses.

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