Finite Math Examples

$\frac{63 x^{5} - 28 x^{3}}{- 7 x}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $7 x^{3}$ out of $63 x^{5} - 28 x^{3}$ .

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Step 1.1.1

Factor $7 x^{3}$ out of $63 x^{5}$ .

$\frac{7 x^{3} (9 x^{2}) - 28 x^{3}}{- 7 x}$

Step 1.1.2

Factor $7 x^{3}$ out of $- 28 x^{3}$ .

$\frac{7 x^{3} (9 x^{2}) + 7 x^{3} (- 4)}{- 7 x}$

Step 1.1.3

Factor $7 x^{3}$ out of $7 x^{3} (9 x^{2}) + 7 x^{3} (- 4)$ .

$\frac{7 x^{3} (9 x^{2} - 4)}{- 7 x}$

Step 1.2

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

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

Step 1.3

Rewrite $4$ as $2^{2}$ .

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

Step 1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 2$ .

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

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $7$ and $- 7$ .

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Step 2.1.1

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

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.2

Simplify terms.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Move the negative one from the denominator of $\frac{(x^{2} (3 x + 2)) (3 x - 2)}{- 1}$ .

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

Step 2.2.2

Rewrite $- 1 \cdot ((x^{2} (3 x + 2)) (3 x - 2))$ as $- ((x^{2} (3 x + 2)) (3 x - 2))$ .

$- ((x^{2} (3 x + 2)) (3 x - 2))$

Step 2.2.3

Apply the distributive property.

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

Step 2.2.4

Reorder.

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Step 2.2.4.1

Rewrite using the commutative property of multiplication.

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

Step 2.2.4.2

Move $2$ to the left of $x^{2}$ .

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

Step 2.3

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

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Step 2.3.1

Move $x$ .

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

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.3

Add $1$ and $2$ .

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

$- ((3 x^{3} + 2 x^{2}) (3 x - 2))$

Step 3

Expand $(3 x^{3} + 2 x^{2}) (3 x - 2)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$- (3 x^{3} (3 x - 2) + 2 x^{2} (3 x - 2))$

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.1.2

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 4.1.2.1

Move $x$ .

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

Step 4.1.2.2

Multiply $x$ by $x^{3}$ .

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Step 4.1.2.2.1

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

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

Step 4.1.2.2.2

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

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

Step 4.1.2.3

Add $1$ and $3$ .

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

Step 4.1.3

Multiply $3$ by $3$ .

$- (9 x^{4} + 3 x^{3} \cdot - 2 + 2 x^{2} (3 x) + 2 x^{2} \cdot - 2)$

Step 4.1.4

Multiply $- 2$ by $3$ .

$- (9 x^{4} - 6 x^{3} + 2 x^{2} (3 x) + 2 x^{2} \cdot - 2)$

Step 4.1.5

Rewrite using the commutative property of multiplication.

$- (9 x^{4} - 6 x^{3} + 2 \cdot 3 x^{2} x + 2 x^{2} \cdot - 2)$

Step 4.1.6

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

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Step 4.1.6.1

Move $x$ .

$- (9 x^{4} - 6 x^{3} + 2 \cdot 3 (x \cdot x^{2}) + 2 x^{2} \cdot - 2)$

Step 4.1.6.2

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

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Step 4.1.6.2.1

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

$- (9 x^{4} - 6 x^{3} + 2 \cdot 3 (x^{1} x^{2}) + 2 x^{2} \cdot - 2)$

Step 4.1.6.2.2

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

$- (9 x^{4} - 6 x^{3} + 2 \cdot 3 x^{1 + 2} + 2 x^{2} \cdot - 2)$

Step 4.1.6.3

Add $1$ and $2$ .

$- (9 x^{4} - 6 x^{3} + 2 \cdot 3 x^{3} + 2 x^{2} \cdot - 2)$

Step 4.1.7

Multiply $2$ by $3$ .

$- (9 x^{4} - 6 x^{3} + 6 x^{3} + 2 x^{2} \cdot - 2)$

Step 4.1.8

Multiply $- 2$ by $2$ .

$- (9 x^{4} - 6 x^{3} + 6 x^{3} - 4 x^{2})$

Step 4.2

Add $- 6 x^{3}$ and $6 x^{3}$ .

$- (9 x^{4} + 0 - 4 x^{2})$

Step 4.3

Add $9 x^{4}$ and $0$ .

$- (9 x^{4} - 4 x^{2})$

Step 5

Apply the distributive property.

$- (9 x^{4}) - (- 4 x^{2})$

Step 6

Multiply.

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Step 6.1

Multiply $9$ by $- 1$ .

$- 9 x^{4} - (- 4 x^{2})$

Step 6.2

Multiply $- 4$ by $- 1$ .

$- 9 x^{4} + 4 x^{2}$