Finite Math Examples

$\frac{8 x^{5} + 36 x^{4} - 20 x^{3}}{4 x^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $4 x^{3}$ out of $8 x^{5} + 36 x^{4} - 20 x^{3}$ .

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Step 1.1.1

Factor $4 x^{3}$ out of $8 x^{5}$ .

$\frac{4 x^{3} (2 x^{2}) + 36 x^{4} - 20 x^{3}}{4 x^{2}}$

Step 1.1.2

Factor $4 x^{3}$ out of $36 x^{4}$ .

$\frac{4 x^{3} (2 x^{2}) + 4 x^{3} (9 x) - 20 x^{3}}{4 x^{2}}$

Step 1.1.3

Factor $4 x^{3}$ out of $- 20 x^{3}$ .

$\frac{4 x^{3} (2 x^{2}) + 4 x^{3} (9 x) + 4 x^{3} (- 5)}{4 x^{2}}$

Step 1.1.4

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

$\frac{4 x^{3} (2 x^{2} + 9 x) + 4 x^{3} (- 5)}{4 x^{2}}$

Step 1.1.5

Factor $4 x^{3}$ out of $4 x^{3} (2 x^{2} + 9 x) + 4 x^{3} (- 5)$ .

$\frac{4 x^{3} (2 x^{2} + 9 x - 5)}{4 x^{2}}$

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 $a \cdot c = 2 \cdot - 5 = - 10$ and whose sum is $b = 9$ .

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Step 1.2.1.1

Factor $9$ out of $9 x$ .

$\frac{4 x^{3} (2 x^{2} + 9 (x) - 5)}{4 x^{2}}$

Step 1.2.1.2

Rewrite $9$ as $- 1$ plus $10$

$\frac{4 x^{3} (2 x^{2} + (- 1 + 10) x - 5)}{4 x^{2}}$

Step 1.2.1.3

Apply the distributive property.

$\frac{4 x^{3} (2 x^{2} - 1 x + 10 x - 5)}{4 x^{2}}$

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.

$\frac{4 x^{3} ((2 x^{2} - 1 x) + 10 x - 5)}{4 x^{2}}$

Step 1.2.2.2

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

$\frac{4 x^{3} (x (2 x - 1) + 5 (2 x - 1))}{4 x^{2}}$

Step 1.2.3

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

$\frac{4 x^{3} ((2 x - 1) (x + 5))}{4 x^{2}}$

$\frac{4 x^{3} (2 x - 1) (x + 5)}{4 x^{2}}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $4$ .

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Step 2.1.1

Cancel the common factor.

$\frac{4 x^{3} (2 x - 1) (x + 5)}{4 x^{2}}$

Step 2.1.2

Rewrite the expression.

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

Step 2.2

Simplify terms.

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Step 2.2.1

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 2.2.1.1

Factor $x^{2}$ out of $(x^{3} (2 x - 1)) (x + 5)$ .

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

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Multiply by $1$ .

$\frac{x^{2} ((x (2 x - 1)) (x + 5))}{x^{2} \cdot 1}$

Step 2.2.1.2.2

Cancel the common factor.

$\frac{x^{2} ((x (2 x - 1)) (x + 5))}{x^{2} \cdot 1}$

Step 2.2.1.2.3

Rewrite the expression.

$\frac{(x (2 x - 1)) (x + 5)}{1}$

Step 2.2.1.2.4

Divide $(x (2 x - 1)) (x + 5)$ by $1$ .

$(x (2 x - 1)) (x + 5)$

Step 2.2.2

Apply the distributive property.

$(x (2 x) + x \cdot - 1) (x + 5)$

Step 2.2.3

Reorder.

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Step 2.2.3.1

Rewrite using the commutative property of multiplication.

$(2 x \cdot x + x \cdot - 1) (x + 5)$

Step 2.2.3.2

Move $- 1$ to the left of $x$ .

$(2 x \cdot x - 1 \cdot x) (x + 5)$

Step 2.3

Simplify each term.

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Step 2.3.1

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

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Step 2.3.1.1

Move $x$ .

$(2 (x \cdot x) - 1 \cdot x) (x + 5)$

Step 2.3.1.2

Multiply $x$ by $x$ .

$(2 x^{2} - 1 \cdot x) (x + 5)$

Step 2.3.2

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

$(2 x^{2} - x) (x + 5)$

Step 3

Expand $(2 x^{2} - x) (x + 5)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$2 x^{2} (x + 5) - x (x + 5)$

Step 3.2

Apply the distributive property.

$2 x^{2} x + 2 x^{2} \cdot 5 - x (x + 5)$

Step 3.3

Apply the distributive property.

$2 x^{2} x + 2 x^{2} \cdot 5 - x \cdot x - x \cdot 5$

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

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

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Step 4.1.1.1

Move $x$ .

$2 (x \cdot x^{2}) + 2 x^{2} \cdot 5 - x \cdot x - x \cdot 5$

Step 4.1.1.2

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

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Step 4.1.1.2.1

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

$2 (x^{1} x^{2}) + 2 x^{2} \cdot 5 - x \cdot x - x \cdot 5$

Step 4.1.1.2.2

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

$2 x^{1 + 2} + 2 x^{2} \cdot 5 - x \cdot x - x \cdot 5$

Step 4.1.1.3

Add $1$ and $2$ .

$2 x^{3} + 2 x^{2} \cdot 5 - x \cdot x - x \cdot 5$

Step 4.1.2

Multiply $5$ by $2$ .

$2 x^{3} + 10 x^{2} - x \cdot x - x \cdot 5$

Step 4.1.3

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

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Step 4.1.3.1

Move $x$ .

$2 x^{3} + 10 x^{2} - (x \cdot x) - x \cdot 5$

Step 4.1.3.2

Multiply $x$ by $x$ .

$2 x^{3} + 10 x^{2} - x^{2} - x \cdot 5$

Step 4.1.4

Multiply $5$ by $- 1$ .

$2 x^{3} + 10 x^{2} - x^{2} - 5 x$

Step 4.2

Subtract $x^{2}$ from $10 x^{2}$ .

$2 x^{3} + 9 x^{2} - 5 x$