Basic Math Examples

$\frac{8 x^{3} + 4 x^{2}}{x^{2} - 3 x + 2} \cdot \frac{x^{2} - 49}{x y^{4}}$

Step 1

Factor $4 x^{2}$ out of $8 x^{3} + 4 x^{2}$ .

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Step 1.1

Factor $4 x^{2}$ out of $8 x^{3}$ .

$\frac{4 x^{2} (2 x) + 4 x^{2}}{x^{2} - 3 x + 2} \cdot \frac{x^{2} - 49}{x y^{4}}$

Step 1.2

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

$\frac{4 x^{2} (2 x) + 4 x^{2} (1)}{x^{2} - 3 x + 2} \cdot \frac{x^{2} - 49}{x y^{4}}$

Step 1.3

Factor $4 x^{2}$ out of $4 x^{2} (2 x) + 4 x^{2} (1)$ .

$\frac{4 x^{2} (2 x + 1)}{x^{2} - 3 x + 2} \cdot \frac{x^{2} - 49}{x y^{4}}$

Step 2

Factor $x^{2} - 3 x + 2$ 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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 2.2

Write the factored form using these integers.

$\frac{4 x^{2} (2 x + 1)}{(x - 2) (x - 1)} \cdot \frac{x^{2} - 49}{x y^{4}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $49$ as $7^{2}$ .

$\frac{4 x^{2} (2 x + 1)}{(x - 2) (x - 1)} \cdot \frac{x^{2} - 7^{2}}{x y^{4}}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$\frac{4 x^{2} (2 x + 1)}{(x - 2) (x - 1)} \cdot \frac{(x + 7) (x - 7)}{x y^{4}}$

Step 4

Simplify terms.

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Step 4.1

Combine.

$\frac{4 x^{2} (2 x + 1) ((x + 7) (x - 7))}{(x - 2) (x - 1) (x y^{4})}$

Step 4.2

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

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Step 4.2.1

Factor $x$ out of $4 x^{2} (2 x + 1) ((x + 7) (x - 7))$ .

$\frac{x (4 x (2 x + 1) ((x + 7) (x - 7)))}{(x - 2) (x - 1) (x y^{4})}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $x$ out of $(x - 2) (x - 1) (x y^{4})$ .

$\frac{x (4 x (2 x + 1) ((x + 7) (x - 7)))}{x ((x - 2) (x - 1) (y^{4}))}$

Step 4.2.2.2

Cancel the common factor.

$\frac{x (4 x (2 x + 1) ((x + 7) (x - 7)))}{x ((x - 2) (x - 1) (y^{4}))}$

Step 4.2.2.3

Rewrite the expression.

$\frac{4 x (2 x + 1) ((x + 7) (x - 7))}{(x - 2) (x - 1) (y^{4})}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite.

$\frac{4 x^{2} (2 x + 1) ((x + 7) (x - 7))}{(x - 2) (x - 1) y^{4}}$

Step 5.2

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

$\frac{4 (x \cdot x) (2 x + 1) ((x + 7) (x - 7))}{(x - 2) (x - 1) y^{4}}$

Step 5.3

Rewrite.

$\frac{4 x^{1} (2 x + 1) ((x + 7) (x - 7))}{(x - 2) (x - 1) y^{4}}$

Step 5.4

Simplify.

$\frac{4 x (2 x + 1) ((x + 7) (x - 7))}{(x - 2) (x - 1) y^{4}}$

Step 5.5

Remove unnecessary parentheses.

$\frac{4 x (2 x + 1) (x + 7) (x - 7)}{(x - 2) (x - 1) y^{4}}$

Step 6

Reorder factors in $\frac{4 x (2 x + 1) (x + 7) (x - 7)}{(x - 2) (x - 1) y^{4}}$ .

$\frac{4 x (2 x + 1) (x + 7) (x - 7)}{y^{4} (x - 2) (x - 1)}$