Pre-Algebra Examples

$\frac{16 x^{5} - 56 x^{4} + 24 x^{3}}{8 x^{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $8 x^{3}$ out of $16 x^{5} - 56 x^{4} + 24 x^{3}$ .

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Step 1.1.1

Factor $8 x^{3}$ out of $16 x^{5}$ .

$\frac{8 x^{3} (2 x^{2}) - 56 x^{4} + 24 x^{3}}{8 x^{4}}$

Step 1.1.2

Factor $8 x^{3}$ out of $- 56 x^{4}$ .

$\frac{8 x^{3} (2 x^{2}) + 8 x^{3} (- 7 x) + 24 x^{3}}{8 x^{4}}$

Step 1.1.3

Factor $8 x^{3}$ out of $24 x^{3}$ .

$\frac{8 x^{3} (2 x^{2}) + 8 x^{3} (- 7 x) + 8 x^{3} (3)}{8 x^{4}}$

Step 1.1.4

Factor $8 x^{3}$ out of $8 x^{3} (2 x^{2}) + 8 x^{3} (- 7 x)$ .

$\frac{8 x^{3} (2 x^{2} - 7 x) + 8 x^{3} (3)}{8 x^{4}}$

Step 1.1.5

Factor $8 x^{3}$ out of $8 x^{3} (2 x^{2} - 7 x) + 8 x^{3} (3)$ .

$\frac{8 x^{3} (2 x^{2} - 7 x + 3)}{8 x^{4}}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

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Step 1.2.1.1

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

$\frac{8 x^{3} (2 x^{2} - 7 (x) + 3)}{8 x^{4}}$

Step 1.2.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

$\frac{8 x^{3} (2 x^{2} + (- 1 - 6) x + 3)}{8 x^{4}}$

Step 1.2.1.3

Apply the distributive property.

$\frac{8 x^{3} (2 x^{2} - 1 x - 6 x + 3)}{8 x^{4}}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{8 x^{3} ((2 x^{2} - 1 x) - 6 x + 3)}{8 x^{4}}$

Step 1.2.2.2

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

$\frac{8 x^{3} (x (2 x - 1) - 3 (2 x - 1))}{8 x^{4}}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{8 x^{3} ((2 x - 1) (x - 3))}{8 x^{4}}$

$\frac{8 x^{3} (2 x - 1) (x - 3)}{8 x^{4}}$

Step 2

Cancel the common factor of $8$ .

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Step 2.1

Cancel the common factor.

$\frac{8 x^{3} (2 x - 1) (x - 3)}{8 x^{4}}$

Step 2.2

Rewrite the expression.

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

Step 3

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 3.1

Factor $x^{3}$ out of $(x^{3} (2 x - 1)) (x - 3)$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Expand $(2 x - 1) (x - 3)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 3 - 1 x - 1 \cdot - 3}{x}$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

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Step 5.1.1.1

Move $x$ .

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

Step 5.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot - 3 - 1 x - 1 \cdot - 3}{x}$

Step 5.1.2

Multiply $- 3$ by $2$ .

$\frac{2 x^{2} - 6 x - 1 x - 1 \cdot - 3}{x}$

Step 5.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{2 x^{2} - 6 x - x - 1 \cdot - 3}{x}$

Step 5.1.4

Multiply $- 1$ by $- 3$ .

$\frac{2 x^{2} - 6 x - x + 3}{x}$

Step 5.2

Subtract $x$ from $- 6 x$ .

$\frac{2 x^{2} - 7 x + 3}{x}$

Step 6

Split the fraction $\frac{2 x^{2} - 7 x + 3}{x}$ into two fractions.

$\frac{2 x^{2} - 7 x}{x} + \frac{3}{x}$

Step 7

Split the fraction $\frac{2 x^{2} - 7 x}{x}$ into two fractions.

$\frac{2 x^{2}}{x} + \frac{- 7 x}{x} + \frac{3}{x}$

Step 8

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

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Step 8.1

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

$\frac{x (2 x)}{x} + \frac{- 7 x}{x} + \frac{3}{x}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

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

$\frac{x (2 x)}{x^{1}} + \frac{- 7 x}{x} + \frac{3}{x}$

Step 8.2.2

Factor $x$ out of $x^{1}$ .

$\frac{x (2 x)}{x \cdot 1} + \frac{- 7 x}{x} + \frac{3}{x}$

Step 8.2.3

Cancel the common factor.

$\frac{x (2 x)}{x \cdot 1} + \frac{- 7 x}{x} + \frac{3}{x}$

Step 8.2.4

Rewrite the expression.

$\frac{2 x}{1} + \frac{- 7 x}{x} + \frac{3}{x}$

Step 8.2.5

Divide $2 x$ by $1$ .

$2 x + \frac{- 7 x}{x} + \frac{3}{x}$

Step 9

Cancel the common factor of $x$ .

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Step 9.1

Cancel the common factor.

$2 x + \frac{- 7 x}{x} + \frac{3}{x}$

Step 9.2

Divide $- 7$ by $1$ .

$2 x - 7 + \frac{3}{x}$