Pre-Algebra Examples

$(30 x^{4} y^{2} - 46 x^{3} y^{2} + 16 x^{2} y^{2}) \div (2 x^{2} y - 2 x y)$

Step 1

Rewrite the division as a fraction.

$\frac{30 x^{4} y^{2} - 46 x^{3} y^{2} + 16 x^{2} y^{2}}{2 x^{2} y - 2 x y}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $2 x^{2} y^{2}$ out of $30 x^{4} y^{2} - 46 x^{3} y^{2} + 16 x^{2} y^{2}$ .

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Step 2.1.1

Factor $2 x^{2} y^{2}$ out of $30 x^{4} y^{2}$ .

$\frac{2 x^{2} y^{2} (15 x^{2}) - 46 x^{3} y^{2} + 16 x^{2} y^{2}}{2 x^{2} y - 2 x y}$

Step 2.1.2

Factor $2 x^{2} y^{2}$ out of $- 46 x^{3} y^{2}$ .

$\frac{2 x^{2} y^{2} (15 x^{2}) + 2 x^{2} y^{2} (- 23 x) + 16 x^{2} y^{2}}{2 x^{2} y - 2 x y}$

Step 2.1.3

Factor $2 x^{2} y^{2}$ out of $16 x^{2} y^{2}$ .

$\frac{2 x^{2} y^{2} (15 x^{2}) + 2 x^{2} y^{2} (- 23 x) + 2 x^{2} y^{2} (8)}{2 x^{2} y - 2 x y}$

Step 2.1.4

Factor $2 x^{2} y^{2}$ out of $2 x^{2} y^{2} (15 x^{2}) + 2 x^{2} y^{2} (- 23 x)$ .

$\frac{2 x^{2} y^{2} (15 x^{2} - 23 x) + 2 x^{2} y^{2} (8)}{2 x^{2} y - 2 x y}$

Step 2.1.5

Factor $2 x^{2} y^{2}$ out of $2 x^{2} y^{2} (15 x^{2} - 23 x) + 2 x^{2} y^{2} (8)$ .

$\frac{2 x^{2} y^{2} (15 x^{2} - 23 x + 8)}{2 x^{2} y - 2 x y}$

Step 2.2

Factor by grouping.

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Step 2.2.1

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 $a \cdot c = 15 \cdot 8 = 120$ and whose sum is $b = - 23$ .

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Step 2.2.1.1

Factor $- 23$ out of $- 23 x$ .

$\frac{2 x^{2} y^{2} (15 x^{2} - 23 (x) + 8)}{2 x^{2} y - 2 x y}$

Step 2.2.1.2

Rewrite $- 23$ as $- 8$ plus $- 15$

$\frac{2 x^{2} y^{2} (15 x^{2} + (- 8 - 15) x + 8)}{2 x^{2} y - 2 x y}$

Step 2.2.1.3

Apply the distributive property.

$\frac{2 x^{2} y^{2} (15 x^{2} - 8 x - 15 x + 8)}{2 x^{2} y - 2 x y}$

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

$\frac{2 x^{2} y^{2} ((15 x^{2} - 8 x) - 15 x + 8)}{2 x^{2} y - 2 x y}$

Step 2.2.2.2

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

$\frac{2 x^{2} y^{2} (x (15 x - 8) - (15 x - 8))}{2 x^{2} y - 2 x y}$

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $15 x - 8$ .

$\frac{2 x^{2} y^{2} ((15 x - 8) (x - 1))}{2 x^{2} y - 2 x y}$

$\frac{2 x^{2} y^{2} (15 x - 8) (x - 1)}{2 x^{2} y - 2 x y}$

Step 3

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

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Step 3.1

Factor $2 x y$ out of $2 x^{2} y$ .

$\frac{2 x^{2} y^{2} (15 x - 8) (x - 1)}{2 x y (x) - 2 x y}$

Step 3.2

Factor $2 x y$ out of $- 2 x y$ .

$\frac{2 x^{2} y^{2} (15 x - 8) (x - 1)}{2 x y (x) + 2 x y (- 1)}$

Step 3.3

Factor $2 x y$ out of $2 x y (x) + 2 x y (- 1)$ .

$\frac{2 x^{2} y^{2} (15 x - 8) (x - 1)}{2 x y (x - 1)}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

$\frac{2 x^{2} y^{2} (15 x - 8) (x - 1)}{2 x y (x - 1)}$

Step 4.2

Rewrite the expression.

$\frac{(x^{2} y^{2} (15 x - 8)) (x - 1)}{x y (x - 1)}$

Step 5

Cancel the common factor of $x^{2}$ and $x$ .

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Step 5.1

Factor $x$ out of $(x^{2} y^{2} (15 x - 8)) (x - 1)$ .

$\frac{x ((x y^{2} (15 x - 8)) (x - 1))}{x y (x - 1)}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $x$ out of $x y (x - 1)$ .

$\frac{x ((x y^{2} (15 x - 8)) (x - 1))}{x (y (x - 1))}$

Step 5.2.2

Cancel the common factor.

$\frac{x ((x y^{2} (15 x - 8)) (x - 1))}{x (y (x - 1))}$

Step 5.2.3

Rewrite the expression.

$\frac{(x y^{2} (15 x - 8)) (x - 1)}{y (x - 1)}$

Step 6

Cancel the common factor of $y^{2}$ and $y$ .

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Step 6.1

Factor $y$ out of $(x y^{2} (15 x - 8)) (x - 1)$ .

$\frac{y (x y (15 x - 8) (x - 1))}{y (x - 1)}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Cancel the common factor.

$\frac{y (x y (15 x - 8) (x - 1))}{y (x - 1)}$

Step 6.2.2

Rewrite the expression.

$\frac{x y (15 x - 8) (x - 1)}{x - 1}$

Step 7

Cancel the common factor of $x - 1$ .

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Step 7.1

Cancel the common factor.

$\frac{x y (15 x - 8) (x - 1)}{x - 1}$

Step 7.2

Divide $x y (15 x - 8)$ by $1$ .

$x y (15 x - 8)$

Step 8

Apply the distributive property.

$x y (15 x) + x y \cdot - 8$

Step 9

Rewrite using the commutative property of multiplication.

$15 x y x + x y \cdot - 8$

Step 10

Move $- 8$ to the left of $x y$ .

$15 x y x - 8 \cdot (x y)$

Step 11

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

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Step 11.1

Move $x$ .

$15 (x \cdot x) y - 8 \cdot (x y)$

Step 11.2

Multiply $x$ by $x$ .

$15 x^{2} y - 8 \cdot (x y)$

$15 x^{2} y - 8 x y$