Algebra Examples

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

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

Expand $(x^{2} + x - 1) (3 x^{2} + 6 x + 6)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.2

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

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Step 3.1.2.1

Move $x^{2}$ .

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Add $2$ and $2$ .

$3 x^{4} + x^{2} (6 x) + x^{2} \cdot 6 + x (3 x^{2}) + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.3

Rewrite using the commutative property of multiplication.

$3 x^{4} + 6 x^{2} x + x^{2} \cdot 6 + x (3 x^{2}) + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.4

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

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Step 3.1.4.1

Move $x$ .

$3 x^{4} + 6 (x \cdot x^{2}) + x^{2} \cdot 6 + x (3 x^{2}) + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.4.2

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

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Step 3.1.4.2.1

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

$3 x^{4} + 6 (x^{1} x^{2}) + x^{2} \cdot 6 + x (3 x^{2}) + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.4.2.2

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

$3 x^{4} + 6 x^{1 + 2} + x^{2} \cdot 6 + x (3 x^{2}) + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.4.3

Add $1$ and $2$ .

$3 x^{4} + 6 x^{3} + x^{2} \cdot 6 + x (3 x^{2}) + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.5

Move $6$ to the left of $x^{2}$ .

$3 x^{4} + 6 x^{3} + 6 \cdot x^{2} + x (3 x^{2}) + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.6

Rewrite using the commutative property of multiplication.

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x \cdot x^{2} + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.7

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

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Step 3.1.7.1

Move $x^{2}$ .

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 (x^{2} x) + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.7.2

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

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Step 3.1.7.2.1

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

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 (x^{2} x^{1}) + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.7.2.2

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

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{2 + 1} + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.7.3

Add $2$ and $1$ .

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + x (6 x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.8

Rewrite using the commutative property of multiplication.

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + 6 x \cdot x + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.9

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

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Step 3.1.9.1

Move $x$ .

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + 6 (x \cdot x) + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.9.2

Multiply $x$ by $x$ .

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + 6 x^{2} + x \cdot 6 - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.10

Move $6$ to the left of $x$ .

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + 6 x^{2} + 6 \cdot x - 1 (3 x^{2}) - 1 (6 x) - 1 \cdot 6$

Step 3.1.11

Multiply $3$ by $- 1$ .

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + 6 x^{2} + 6 x - 3 x^{2} - 1 (6 x) - 1 \cdot 6$

Step 3.1.12

Multiply $6$ by $- 1$ .

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

Step 3.1.13

Multiply $- 1$ by $6$ .

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + 6 x^{2} + 6 x - 3 x^{2} - 6 x - 6$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Combine the opposite terms in $3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + 6 x^{2} + 6 x - 3 x^{2} - 6 x - 6$ .

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Step 3.2.1.1

Subtract $6 x$ from $6 x$ .

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + 6 x^{2} - 3 x^{2} + 0 - 6$

Step 3.2.1.2

Add $3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + 6 x^{2} - 3 x^{2}$ and $0$ .

$3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x^{3} + 6 x^{2} - 3 x^{2} - 6$

Step 3.2.2

Add $6 x^{3}$ and $3 x^{3}$ .

$3 x^{4} + 9 x^{3} + 6 x^{2} + 6 x^{2} - 3 x^{2} - 6$

Step 3.2.3

Add $6 x^{2}$ and $6 x^{2}$ .

$3 x^{4} + 9 x^{3} + 12 x^{2} - 3 x^{2} - 6$

Step 3.2.4

Subtract $3 x^{2}$ from $12 x^{2}$ .

$3 x^{4} + 9 x^{3} + 9 x^{2} - 6$