Algebra Examples

$(6 n^{2} - 6 n - 5) (7 n^{2} + 6 n - 5)$

Step 1

Expand $(6 n^{2} - 6 n - 5) (7 n^{2} + 6 n - 5)$ by multiplying each term in the first expression by each term in the second expression.

$6 n^{2} (7 n^{2}) + 6 n^{2} (6 n) + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$6 \cdot 7 n^{2} n^{2} + 6 n^{2} (6 n) + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.2

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

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Step 2.1.2.1

Move $n^{2}$ .

$6 \cdot 7 (n^{2} n^{2}) + 6 n^{2} (6 n) + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.2.2

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

$6 \cdot 7 n^{2 + 2} + 6 n^{2} (6 n) + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.2.3

Add $2$ and $2$ .

$6 \cdot 7 n^{4} + 6 n^{2} (6 n) + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.3

Multiply $6$ by $7$ .

$42 n^{4} + 6 n^{2} (6 n) + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$42 n^{4} + 6 \cdot 6 n^{2} n + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.5

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

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Step 2.1.5.1

Move $n$ .

$42 n^{4} + 6 \cdot 6 (n \cdot n^{2}) + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.5.2

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

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Step 2.1.5.2.1

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

$42 n^{4} + 6 \cdot 6 (n^{1} n^{2}) + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.5.2.2

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

$42 n^{4} + 6 \cdot 6 n^{1 + 2} + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.5.3

Add $1$ and $2$ .

$42 n^{4} + 6 \cdot 6 n^{3} + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.6

Multiply $6$ by $6$ .

$42 n^{4} + 36 n^{3} + 6 n^{2} \cdot - 5 - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.7

Multiply $- 5$ by $6$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 6 n (7 n^{2}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.8

Rewrite using the commutative property of multiplication.

$42 n^{4} + 36 n^{3} - 30 n^{2} - 6 \cdot 7 n \cdot n^{2} - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.9

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

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Step 2.1.9.1

Move $n^{2}$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 6 \cdot 7 (n^{2} n) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.9.2

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

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Step 2.1.9.2.1

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

$42 n^{4} + 36 n^{3} - 30 n^{2} - 6 \cdot 7 (n^{2} n^{1}) - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.9.2.2

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

$42 n^{4} + 36 n^{3} - 30 n^{2} - 6 \cdot 7 n^{2 + 1} - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.9.3

Add $2$ and $1$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 6 \cdot 7 n^{3} - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.10

Multiply $- 6$ by $7$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 6 n (6 n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.11

Rewrite using the commutative property of multiplication.

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 6 \cdot 6 n \cdot n - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.12

Multiply $n$ by $n$ by adding the exponents.

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Step 2.1.12.1

Move $n$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 6 \cdot 6 (n \cdot n) - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.12.2

Multiply $n$ by $n$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 6 \cdot 6 n^{2} - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.13

Multiply $- 6$ by $6$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 36 n^{2} - 6 n \cdot - 5 - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.14

Multiply $- 5$ by $- 6$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 36 n^{2} + 30 n - 5 (7 n^{2}) - 5 (6 n) - 5 \cdot - 5$

Step 2.1.15

Multiply $7$ by $- 5$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 36 n^{2} + 30 n - 35 n^{2} - 5 (6 n) - 5 \cdot - 5$

Step 2.1.16

Multiply $6$ by $- 5$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 36 n^{2} + 30 n - 35 n^{2} - 30 n - 5 \cdot - 5$

Step 2.1.17

Multiply $- 5$ by $- 5$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 36 n^{2} + 30 n - 35 n^{2} - 30 n + 25$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Combine the opposite terms in $42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 36 n^{2} + 30 n - 35 n^{2} - 30 n + 25$ .

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Step 2.2.1.1

Subtract $30 n$ from $30 n$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 36 n^{2} - 35 n^{2} + 0 + 25$

Step 2.2.1.2

Add $42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 36 n^{2} - 35 n^{2}$ and $0$ .

$42 n^{4} + 36 n^{3} - 30 n^{2} - 42 n^{3} - 36 n^{2} - 35 n^{2} + 25$

Step 2.2.2

Subtract $42 n^{3}$ from $36 n^{3}$ .

$42 n^{4} - 6 n^{3} - 30 n^{2} - 36 n^{2} - 35 n^{2} + 25$

Step 2.2.3

Subtract $36 n^{2}$ from $- 30 n^{2}$ .

$42 n^{4} - 6 n^{3} - 66 n^{2} - 35 n^{2} + 25$

Step 2.2.4

Subtract $35 n^{2}$ from $- 66 n^{2}$ .

$42 n^{4} - 6 n^{3} - 101 n^{2} + 25$