Basic Math Examples

$\frac{2 (w^{2} + w - 6)}{w^{2} + 4 w - 5} \cdot \frac{w^{3} - 3 w^{2} + 2 w}{4 w^{2} - 6 w}$

Step 1

Factor $w^{2} + w - 6$ 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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 1.2

Write the factored form using these integers.

$\frac{2 ((w - 2) (w + 3))}{w^{2} + 4 w - 5} \cdot \frac{w^{3} - 3 w^{2} + 2 w}{4 w^{2} - 6 w}$

$\frac{2 (w - 2) (w + 3)}{w^{2} + 4 w - 5} \cdot \frac{w^{3} - 3 w^{2} + 2 w}{4 w^{2} - 6 w}$

Step 2

Factor $w^{2} + 4 w - 5$ using the AC method.

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Step 2.1

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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 2.2

Write the factored form using these integers.

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w^{3} - 3 w^{2} + 2 w}{4 w^{2} - 6 w}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $w$ out of $w^{3} - 3 w^{2} + 2 w$ .

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Step 3.1.1

Factor $w$ out of $w^{3}$ .

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w \cdot w^{2} - 3 w^{2} + 2 w}{4 w^{2} - 6 w}$

Step 3.1.2

Factor $w$ out of $- 3 w^{2}$ .

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w \cdot w^{2} + w (- 3 w) + 2 w}{4 w^{2} - 6 w}$

Step 3.1.3

Factor $w$ out of $2 w$ .

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w \cdot w^{2} + w (- 3 w) + w \cdot 2}{4 w^{2} - 6 w}$

Step 3.1.4

Factor $w$ out of $w \cdot w^{2} + w (- 3 w)$ .

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w (w^{2} - 3 w) + w \cdot 2}{4 w^{2} - 6 w}$

Step 3.1.5

Factor $w$ out of $w (w^{2} - 3 w) + w \cdot 2$ .

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w (w^{2} - 3 w + 2)}{4 w^{2} - 6 w}$

Step 3.2

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

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Step 3.2.1

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 3.2.2

Write the factored form using these integers.

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w ((w - 2) (w - 1))}{4 w^{2} - 6 w}$

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w (w - 2) (w - 1)}{4 w^{2} - 6 w}$

Step 4

Simplify terms.

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Step 4.1

Factor $2 w$ out of $4 w^{2} - 6 w$ .

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Step 4.1.1

Factor $2 w$ out of $4 w^{2}$ .

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w (w - 2) (w - 1)}{2 w (2 w) - 6 w}$

Step 4.1.2

Factor $2 w$ out of $- 6 w$ .

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w (w - 2) (w - 1)}{2 w (2 w) + 2 w (- 3)}$

Step 4.1.3

Factor $2 w$ out of $2 w (2 w) + 2 w (- 3)$ .

$\frac{2 (w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w (w - 2) (w - 1)}{2 w (2 w - 3)}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $2 (w - 2) (w + 3)$ .

$\frac{2 ((w - 2) (w + 3))}{(w - 1) (w + 5)} \cdot \frac{w (w - 2) (w - 1)}{2 w (2 w - 3)}$

Step 4.2.2

Factor $2$ out of $2 w (2 w - 3)$ .

$\frac{2 ((w - 2) (w + 3))}{(w - 1) (w + 5)} \cdot \frac{w (w - 2) (w - 1)}{2 (w (2 w - 3))}$

Step 4.2.3

Cancel the common factor.

$\frac{2 ((w - 2) (w + 3))}{(w - 1) (w + 5)} \cdot \frac{w (w - 2) (w - 1)}{2 (w (2 w - 3))}$

Step 4.2.4

Rewrite the expression.

$\frac{(w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{w (w - 2) (w - 1)}{w (2 w - 3)}$

Step 4.3

Cancel the common factor of $w - 1$ .

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Step 4.3.1

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

$\frac{(w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{(w - 1) (w (w - 2))}{w (2 w - 3)}$

Step 4.3.2

Cancel the common factor.

$\frac{(w - 2) (w + 3)}{(w - 1) (w + 5)} \cdot \frac{(w - 1) (w (w - 2))}{w (2 w - 3)}$

Step 4.3.3

Rewrite the expression.

$\frac{(w - 2) (w + 3)}{w + 5} \cdot \frac{w (w - 2)}{w (2 w - 3)}$

Step 4.4

Multiply $\frac{(w - 2) (w + 3)}{w + 5}$ by $\frac{w (w - 2)}{w (2 w - 3)}$ .

$\frac{(w - 2) (w + 3) (w (w - 2))}{(w + 5) (w (2 w - 3))}$

Step 5

Raise $w - 2$ to the power of $1$ .

$\frac{(w + 3) (w ({(w - 2)}^{1} (w - 2)))}{(w + 5) (w (2 w - 3))}$

Step 6

Raise $w - 2$ to the power of $1$ .

$\frac{(w + 3) (w ({(w - 2)}^{1} {(w - 2)}^{1}))}{(w + 5) (w (2 w - 3))}$

Step 7

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

$\frac{(w + 3) (w {(w - 2)}^{1 + 1})}{(w + 5) (w (2 w - 3))}$

Step 8

Add $1$ and $1$ .

$\frac{(w + 3) (w {(w - 2)}^{2})}{(w + 5) (w (2 w - 3))}$

Step 9

Cancel the common factor of $w$ .

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Step 9.1

Cancel the common factor.

$\frac{(w + 3) (w {(w - 2)}^{2})}{(w + 5) (w (2 w - 3))}$

Step 9.2

Rewrite the expression.

$\frac{(w + 3) ({(w - 2)}^{2})}{(w + 5) (2 w - 3)}$