Pre-Algebra Examples

$(27 x^{5} + 9 x^{4} - 18 x^{3}) \div 9 x^{2}$

Step 1

Rewrite the division as a fraction.

$\frac{27 x^{5} + 9 x^{4} - 18 x^{3}}{9 x^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $9 x^{3}$ out of $27 x^{5} + 9 x^{4} - 18 x^{3}$ .

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Step 2.1.1

Factor $9 x^{3}$ out of $27 x^{5}$ .

$\frac{9 x^{3} (3 x^{2}) + 9 x^{4} - 18 x^{3}}{9 x^{2}}$

Step 2.1.2

Factor $9 x^{3}$ out of $9 x^{4}$ .

$\frac{9 x^{3} (3 x^{2}) + 9 x^{3} (x) - 18 x^{3}}{9 x^{2}}$

Step 2.1.3

Factor $9 x^{3}$ out of $- 18 x^{3}$ .

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.2

Factor by grouping.

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Step 2.2.1

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 1$ .

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Step 2.2.1.1

Multiply by $1$ .

$\frac{9 x^{3} (3 x^{2} + 1 x - 2)}{9 x^{2}}$

Step 2.2.1.2

Rewrite $1$ as $- 2$ plus $3$

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

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

Step 2.2.2.2

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

$\frac{9 x^{3} (x (3 x - 2) + 1 (3 x - 2))}{9 x^{2}}$

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$\frac{9 x^{3} ((3 x - 2) (x + 1))}{9 x^{2}}$

$\frac{9 x^{3} (3 x - 2) (x + 1)}{9 x^{2}}$

Step 3

Cancel the common factor of $9$ .

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Step 3.1

Cancel the common factor.

$\frac{9 x^{3} (3 x - 2) (x + 1)}{9 x^{2}}$

Step 3.2

Rewrite the expression.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.2.4

Divide $(x (3 x - 2)) (x + 1)$ by $1$ .

$(x (3 x - 2)) (x + 1)$

Step 5

Apply the distributive property.

$(x (3 x) + x \cdot - 2) (x + 1)$

Step 6

Rewrite using the commutative property of multiplication.

$(3 x \cdot x + x \cdot - 2) (x + 1)$

Step 7

Move $- 2$ to the left of $x$ .

$(3 x \cdot x - 2 \cdot x) (x + 1)$

Step 8

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

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Step 8.1

Move $x$ .

$(3 (x \cdot x) - 2 \cdot x) (x + 1)$

Step 8.2

Multiply $x$ by $x$ .

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

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

Step 9

Expand $(3 x^{2} - 2 x) (x + 1)$ using the FOIL Method.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Apply the distributive property.

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

Step 10

Simplify and combine like terms.

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Step 10.1

Simplify each term.

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Step 10.1.1

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

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Step 10.1.1.1

Move $x$ .

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

Step 10.1.1.2

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

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Step 10.1.1.2.1

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

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

Step 10.1.1.2.2

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

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

Step 10.1.1.3

Add $1$ and $2$ .

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

Step 10.1.2

Multiply $3$ by $1$ .

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

Step 10.1.3

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

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Step 10.1.3.1

Move $x$ .

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

Step 10.1.3.2

Multiply $x$ by $x$ .

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

Step 10.1.4

Multiply $- 2$ by $1$ .

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

Step 10.2

Subtract $2 x^{2}$ from $3 x^{2}$ .

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