Algebra Examples

$\frac{x^{2} - 3 x + 2}{- 4 y} \cdot \frac{2 x - 2}{- x y^{2} + 2 y^{2}}$

Step 1

Factor $x^{2} - 3 x + 2$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 2) (x - 1)}{- 4 y} \cdot \frac{2 x - 2}{- x y^{2} + 2 y^{2}}$

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Factor $2$ out of $2 x$ .

$\frac{(x - 2) (x - 1)}{- 4 y} \cdot \frac{2 (x) - 2}{- x y^{2} + 2 y^{2}}$

Step 2.1.2

Factor $2$ out of $- 2$ .

$\frac{(x - 2) (x - 1)}{- 4 y} \cdot \frac{2 (x) + 2 (- 1)}{- x y^{2} + 2 y^{2}}$

Step 2.1.3

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

$\frac{(x - 2) (x - 1)}{- 4 y} \cdot \frac{2 (x - 1)}{- x y^{2} + 2 y^{2}}$

Step 2.2

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

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Step 2.2.1

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

$\frac{(x - 2) (x - 1)}{- 4 y} \cdot \frac{2 (x - 1)}{y^{2} (- x) + 2 y^{2}}$

Step 2.2.2

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

$\frac{(x - 2) (x - 1)}{- 4 y} \cdot \frac{2 (x - 1)}{y^{2} (- x) + y^{2} \cdot 2}$

Step 2.2.3

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

$\frac{(x - 2) (x - 1)}{- 4 y} \cdot \frac{2 (x - 1)}{y^{2} (- x + 2)}$

Step 2.3

Combine.

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

Step 3

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

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Step 3.1

Move $y^{2}$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

Add $2$ and $1$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $2$ and $- 4$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $- 4 y^{3} (- x + 2)$ .

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

Step 4.1.2.2

Cancel the common factor.

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

Step 4.1.2.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 2$ and $- x + 2$ .

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Step 4.2.1

Factor $- 1$ out of $x$ .

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

Step 4.2.2

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 4.2.3

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

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

Step 4.2.4

Cancel the common factor.

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

Step 4.2.5

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

Dividing two negative values results in a positive value.

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