Pre-Algebra Examples

$(15 a^{6} - 9 a^{8} + 6 a^{4}) \div 3 a^{6}$

Step 1

Rewrite the division as a fraction.

$\frac{15 a^{6} - 9 a^{8} + 6 a^{4}}{3 a^{6}}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $3 a^{4}$ out of $15 a^{6} - 9 a^{8} + 6 a^{4}$ .

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Step 2.1.1

Factor $3 a^{4}$ out of $15 a^{6}$ .

$\frac{3 a^{4} (5 a^{2}) - 9 a^{8} + 6 a^{4}}{3 a^{6}}$

Step 2.1.2

Factor $3 a^{4}$ out of $- 9 a^{8}$ .

$\frac{3 a^{4} (5 a^{2}) + 3 a^{4} (- 3 a^{4}) + 6 a^{4}}{3 a^{6}}$

Step 2.1.3

Factor $3 a^{4}$ out of $6 a^{4}$ .

$\frac{3 a^{4} (5 a^{2}) + 3 a^{4} (- 3 a^{4}) + 3 a^{4} (2)}{3 a^{6}}$

Step 2.1.4

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

$\frac{3 a^{4} (5 a^{2} - 3 a^{4}) + 3 a^{4} (2)}{3 a^{6}}$

Step 2.1.5

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

$\frac{3 a^{4} (5 a^{2} - 3 a^{4} + 2)}{3 a^{6}}$

Step 2.2

Factor by grouping.

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Step 2.2.1

Reorder terms.

$\frac{3 a^{4} (- 3 a^{4} + 5 a^{2} + 2)}{3 a^{6}}$

Step 2.2.2

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 $b = 5$ .

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Step 2.2.2.1

Factor $5$ out of $5 a^{2}$ .

$\frac{3 a^{4} (- 3 a^{4} + 5 (a^{2}) + 2)}{3 a^{6}}$

Step 2.2.2.2

Rewrite $5$ as $- 1$ plus $6$

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

Step 2.2.2.3

Apply the distributive property.

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

Step 2.2.3

Factor out the greatest common factor from each group.

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Step 2.2.3.1

Group the first two terms and the last two terms.

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

Step 2.2.3.2

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

$\frac{3 a^{4} (a^{2} (- 3 a^{2} - 1) - 2 (- 3 a^{2} - 1))}{3 a^{6}}$

Step 2.2.4

Factor the polynomial by factoring out the greatest common factor, $- 3 a^{2} - 1$ .

$\frac{3 a^{4} ((- 3 a^{2} - 1) (a^{2} - 2))}{3 a^{6}}$

$\frac{3 a^{4} (- 3 a^{2} - 1) (a^{2} - 2)}{3 a^{6}}$

Step 3

Cancel the common factor of $3$ .

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Step 3.1

Cancel the common factor.

$\frac{3 a^{4} (- 3 a^{2} - 1) (a^{2} - 2)}{3 a^{6}}$

Step 3.2

Rewrite the expression.

$\frac{(a^{4} (- 3 a^{2} - 1)) (a^{2} - 2)}{a^{6}}$

Step 4

Cancel the common factor of $a^{4}$ and $a^{6}$ .

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Step 4.1

Factor $a^{4}$ out of $(a^{4} (- 3 a^{2} - 1)) (a^{2} - 2)$ .

$\frac{a^{4} ((- 3 a^{2} - 1) (a^{2} - 2))}{a^{6}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $a^{4}$ out of $a^{6}$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Factor $- 1$ out of $- 3 a^{2}$ .

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

Step 6

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 7

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

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

Step 8

Rewrite $- (3 a^{2} + 1)$ as $- 1 (3 a^{2} + 1)$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

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

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Step 10.1

Apply the distributive property.

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

Step 10.2

Apply the distributive property.

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

Step 10.3

Apply the distributive property.

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

Step 11

Simplify and combine like terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

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

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Step 11.1.1.1

Move $a^{2}$ .

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

Step 11.1.1.2

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

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

Step 11.1.1.3

Add $2$ and $2$ .

$- \frac{3 a^{4} + 3 a^{2} \cdot - 2 + 1 a^{2} + 1 \cdot - 2}{a^{2}}$

Step 11.1.2

Multiply $- 2$ by $3$ .

$- \frac{3 a^{4} - 6 a^{2} + 1 a^{2} + 1 \cdot - 2}{a^{2}}$

Step 11.1.3

Multiply $a^{2}$ by $1$ .

$- \frac{3 a^{4} - 6 a^{2} + a^{2} + 1 \cdot - 2}{a^{2}}$

Step 11.1.4

Multiply $- 2$ by $1$ .

$- \frac{3 a^{4} - 6 a^{2} + a^{2} - 2}{a^{2}}$

Step 11.2

Add $- 6 a^{2}$ and $a^{2}$ .

$- \frac{3 a^{4} - 5 a^{2} - 2}{a^{2}}$

Step 12

Split the fraction $\frac{3 a^{4} - 5 a^{2} - 2}{a^{2}}$ into two fractions.

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

Step 13

Split the fraction $\frac{3 a^{4} - 5 a^{2}}{a^{2}}$ into two fractions.

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

Step 14

Cancel the common factor of $a^{4}$ and $a^{2}$ .

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Step 14.1

Factor $a^{2}$ out of $3 a^{4}$ .

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

Step 14.2

Cancel the common factors.

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Step 14.2.1

Multiply by $1$ .

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

Step 14.2.2

Cancel the common factor.

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

Step 14.2.3

Rewrite the expression.

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

Step 14.2.4

Divide $3 a^{2}$ by $1$ .

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

Step 15

Cancel the common factor of $a^{2}$ .

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Step 15.1

Cancel the common factor.

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

Step 15.2

Divide $- 5$ by $1$ .

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

Step 16

Move the negative in front of the fraction.

$- (3 a^{2} - 5 - \frac{2}{a^{2}})$