Pre-Algebra Examples

$2 y^{6} - 5 y^{3} + 5$ , $- (8 y^{6} + 15 y^{3} + 6)$

Step 1

Regroup terms.

$2 y^{6} - (8 y^{6} + 15 y^{3} + 6) - 5 y^{3} + 5$

Step 2

Apply the distributive property.

$2 y^{6} - (8 y^{6}) - (15 y^{3}) - 1 \cdot 6 - 5 y^{3} + 5$

Step 3

Simplify.

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Step 3.1

Multiply $8$ by $- 1$ .

$2 y^{6} - 8 y^{6} - (15 y^{3}) - 1 \cdot 6 - 5 y^{3} + 5$

Step 3.2

Multiply $15$ by $- 1$ .

$2 y^{6} - 8 y^{6} - 15 y^{3} - 1 \cdot 6 - 5 y^{3} + 5$

Step 3.3

Multiply $- 1$ by $6$ .

$2 y^{6} - 8 y^{6} - 15 y^{3} - 6 - 5 y^{3} + 5$

Step 4

Subtract $8 y^{6}$ from $2 y^{6}$ .

$- 6 y^{6} - 15 y^{3} - 6 - 5 y^{3} + 5$

Step 5

Rewrite $- 6 y^{6} - 15 y^{3} - 6$ in a factored form.

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Step 5.1

Factor $- 3$ out of $- 6 y^{6} - 15 y^{3} - 6$ .

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Step 5.1.1

Factor $- 3$ out of $- 6 y^{6}$ .

$- 3 (2 y^{6}) - 15 y^{3} - 6 - 5 y^{3} + 5$

Step 5.1.2

Factor $- 3$ out of $- 15 y^{3}$ .

$- 3 (2 y^{6}) - 3 (5 y^{3}) - 6 - 5 y^{3} + 5$

Step 5.1.3

Factor $- 3$ out of $- 6$ .

$- 3 (2 y^{6}) - 3 (5 y^{3}) - 3 (2) - 5 y^{3} + 5$

Step 5.1.4

Factor $- 3$ out of $- 3 (2 y^{6}) - 3 (5 y^{3})$ .

$- 3 (2 y^{6} + 5 y^{3}) - 3 (2) - 5 y^{3} + 5$

Step 5.1.5

Factor $- 3$ out of $- 3 (2 y^{6} + 5 y^{3}) - 3 (2)$ .

$- 3 (2 y^{6} + 5 y^{3} + 2) - 5 y^{3} + 5$

Step 5.2

Rewrite $y^{6}$ as ${(y^{3})}^{2}$ .

$- 3 (2 {(y^{3})}^{2} + 5 y^{3} + 2) - 5 y^{3} + 5$

Step 5.3

Let $u = y^{3}$ . Substitute $u$ for all occurrences of $y^{3}$ .

$- 3 (2 u^{2} + 5 u + 2) - 5 y^{3} + 5$

Step 5.4

Factor by grouping.

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Step 5.4.1

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = 5$ .

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Step 5.4.1.1

Factor $5$ out of $5 u$ .

$- 3 (2 u^{2} + 5 (u) + 2) - 5 y^{3} + 5$

Step 5.4.1.2

Rewrite $5$ as $1$ plus $4$

$- 3 (2 u^{2} + (1 + 4) u + 2) - 5 y^{3} + 5$

Step 5.4.1.3

Apply the distributive property.

$- 3 (2 u^{2} + 1 u + 4 u + 2) - 5 y^{3} + 5$

Step 5.4.1.4

Multiply $u$ by $1$ .

$- 3 (2 u^{2} + u + 4 u + 2) - 5 y^{3} + 5$

Step 5.4.2

Factor out the greatest common factor from each group.

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Step 5.4.2.1

Group the first two terms and the last two terms.

$- 3 ((2 u^{2} + u) + 4 u + 2) - 5 y^{3} + 5$

Step 5.4.2.2

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

$- 3 (u (2 u + 1) + 2 (2 u + 1)) - 5 y^{3} + 5$

Step 5.4.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$- 3 ((2 u + 1) (u + 2)) - 5 y^{3} + 5$

Step 5.5

Factor.

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Step 5.5.1

Replace all occurrences of $u$ with $y^{3}$ .

$- 3 ((2 y^{3} + 1) (y^{3} + 2)) - 5 y^{3} + 5$

Step 5.5.2

Remove unnecessary parentheses.

$- 3 (2 y^{3} + 1) (y^{3} + 2) - 5 y^{3} + 5$

Step 6

Factor $- 5$ out of $- 5 y^{3} + 5$ .

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Step 6.1

Factor $- 5$ out of $- 5 y^{3}$ .

$- 3 (2 y^{3} + 1) (y^{3} + 2) - 5 (y^{3}) + 5$

Step 6.2

Factor $- 5$ out of $5$ .

$- 3 (2 y^{3} + 1) (y^{3} + 2) - 5 (y^{3}) - 5 (- 1)$

Step 6.3

Factor $- 5$ out of $- 5 (y^{3}) - 5 (- 1)$ .

$- 3 (2 y^{3} + 1) (y^{3} + 2) - 5 (y^{3} - 1)$

Step 7

Rewrite $1$ as $1^{3}$ .

$- 3 (2 y^{3} + 1) (y^{3} + 2) - 5 (y^{3} - 1^{3})$

Step 8

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = 1$ .

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

Step 9

Factor.

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Step 9.1

Simplify.

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Step 9.1.1

Multiply $y$ by $1$ .

$- 3 (2 y^{3} + 1) (y^{3} + 2) - 5 ((y - 1) (y^{2} + y + 1^{2}))$

Step 9.1.2

One to any power is one.

$- 3 (2 y^{3} + 1) (y^{3} + 2) - 5 ((y - 1) (y^{2} + y + 1))$

Step 9.2

Remove unnecessary parentheses.

$- 3 (2 y^{3} + 1) (y^{3} + 2) - 5 (y - 1) (y^{2} + y + 1)$

Step 10

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

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Step 10.1

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

$- (3 (2 y^{3} + 1) (y^{3} + 2)) - 5 (y - 1) (y^{2} + y + 1)$

Step 10.2

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

$- (3 (2 y^{3} + 1) (y^{3} + 2)) - (5 (y - 1) (y^{2} + y + 1))$

Step 10.3

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

$- (3 (2 y^{3} + 1) (y^{3} + 2) + 5 (y - 1) (y^{2} + y + 1))$

Step 11

Apply the distributive property.

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

Step 12

Multiply $2$ by $3$ .

$- ((6 y^{3} + 3 \cdot 1) (y^{3} + 2) + 5 (y - 1) (y^{2} + y + 1))$

Step 13

Multiply $3$ by $1$ .

$- ((6 y^{3} + 3) (y^{3} + 2) + 5 (y - 1) (y^{2} + y + 1))$

Step 14

Expand $(6 y^{3} + 3) (y^{3} + 2)$ using the FOIL Method.

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Step 14.1

Apply the distributive property.

$- (6 y^{3} (y^{3} + 2) + 3 (y^{3} + 2) + 5 (y - 1) (y^{2} + y + 1))$

Step 14.2

Apply the distributive property.

$- (6 y^{3} y^{3} + 6 y^{3} \cdot 2 + 3 (y^{3} + 2) + 5 (y - 1) (y^{2} + y + 1))$

Step 14.3

Apply the distributive property.

$- (6 y^{3} y^{3} + 6 y^{3} \cdot 2 + 3 y^{3} + 3 \cdot 2 + 5 (y - 1) (y^{2} + y + 1))$

Step 15

Simplify and combine like terms.

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Step 15.1

Simplify each term.

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Step 15.1.1

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

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Step 15.1.1.1

Move $y^{3}$ .

$- (6 (y^{3} y^{3}) + 6 y^{3} \cdot 2 + 3 y^{3} + 3 \cdot 2 + 5 (y - 1) (y^{2} + y + 1))$

Step 15.1.1.2

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

$- (6 y^{3 + 3} + 6 y^{3} \cdot 2 + 3 y^{3} + 3 \cdot 2 + 5 (y - 1) (y^{2} + y + 1))$

Step 15.1.1.3

Add $3$ and $3$ .

$- (6 y^{6} + 6 y^{3} \cdot 2 + 3 y^{3} + 3 \cdot 2 + 5 (y - 1) (y^{2} + y + 1))$

Step 15.1.2

Multiply $2$ by $6$ .

$- (6 y^{6} + 12 y^{3} + 3 y^{3} + 3 \cdot 2 + 5 (y - 1) (y^{2} + y + 1))$

Step 15.1.3

Multiply $3$ by $2$ .

$- (6 y^{6} + 12 y^{3} + 3 y^{3} + 6 + 5 (y - 1) (y^{2} + y + 1))$

Step 15.2

Add $12 y^{3}$ and $3 y^{3}$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 (y - 1) (y^{2} + y + 1))$

Step 16

Apply the distributive property.

$- (6 y^{6} + 15 y^{3} + 6 + (5 y + 5 \cdot - 1) (y^{2} + y + 1))$

Step 17

Multiply $5$ by $- 1$ .

$- (6 y^{6} + 15 y^{3} + 6 + (5 y - 5) (y^{2} + y + 1))$

Step 18

Expand $(5 y - 5) (y^{2} + y + 1)$ by multiplying each term in the first expression by each term in the second expression.

$- (6 y^{6} + 15 y^{3} + 6 + 5 y \cdot y^{2} + 5 y \cdot y + 5 y \cdot 1 - 5 y^{2} - 5 y - 5 \cdot 1)$

Step 19

Simplify each term.

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Step 19.1

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

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Step 19.1.1

Move $y^{2}$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 (y^{2} y) + 5 y \cdot y + 5 y \cdot 1 - 5 y^{2} - 5 y - 5 \cdot 1)$

Step 19.1.2

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

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Step 19.1.2.1

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

$- (6 y^{6} + 15 y^{3} + 6 + 5 (y^{2} y^{1}) + 5 y \cdot y + 5 y \cdot 1 - 5 y^{2} - 5 y - 5 \cdot 1)$

Step 19.1.2.2

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

$- (6 y^{6} + 15 y^{3} + 6 + 5 y^{2 + 1} + 5 y \cdot y + 5 y \cdot 1 - 5 y^{2} - 5 y - 5 \cdot 1)$

Step 19.1.3

Add $2$ and $1$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 y^{3} + 5 y \cdot y + 5 y \cdot 1 - 5 y^{2} - 5 y - 5 \cdot 1)$

Step 19.2

Multiply $y$ by $y$ by adding the exponents.

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Step 19.2.1

Move $y$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 y^{3} + 5 (y \cdot y) + 5 y \cdot 1 - 5 y^{2} - 5 y - 5 \cdot 1)$

Step 19.2.2

Multiply $y$ by $y$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 y^{3} + 5 y^{2} + 5 y \cdot 1 - 5 y^{2} - 5 y - 5 \cdot 1)$

Step 19.3

Multiply $5$ by $1$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 y^{3} + 5 y^{2} + 5 y - 5 y^{2} - 5 y - 5 \cdot 1)$

Step 19.4

Multiply $- 5$ by $1$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 y^{3} + 5 y^{2} + 5 y - 5 y^{2} - 5 y - 5)$

Step 20

Combine the opposite terms in $5 y^{3} + 5 y^{2} + 5 y - 5 y^{2} - 5 y - 5$ .

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Step 20.1

Subtract $5 y^{2}$ from $5 y^{2}$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 y^{3} + 5 y + 0 - 5 y - 5)$

Step 20.2

Add $5 y^{3} + 5 y$ and $0$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 y^{3} + 5 y - 5 y - 5)$

Step 20.3

Subtract $5 y$ from $5 y$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 y^{3} + 0 - 5)$

Step 20.4

Add $5 y^{3}$ and $0$ .

$- (6 y^{6} + 15 y^{3} + 6 + 5 y^{3} - 5)$

Step 21

Add $15 y^{3}$ and $5 y^{3}$ .

$- (6 y^{6} + 20 y^{3} + 6 - 5)$

Step 22

Subtract $5$ from $6$ .

$- (6 y^{6} + 20 y^{3} + 1)$

Step 23

The greatest common factor $GCF$ is the term in front of the factored expression.

$- 1$