Algebra Examples

$(2 x^{2} - 7 x - 1) (3 x^{2} + x - 1)$

Step 1

Expand $(2 x^{2} - 7 x - 1) (3 x^{2} + x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$2 x^{2} (3 x^{2}) + 2 x^{2} x + 2 x^{2} \cdot - 1 - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

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.

$2 \cdot 3 x^{2} x^{2} + 2 x^{2} x + 2 x^{2} \cdot - 1 - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.2

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

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Step 2.1.2.1

Move $x^{2}$ .

$2 \cdot 3 (x^{2} x^{2}) + 2 x^{2} x + 2 x^{2} \cdot - 1 - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.2.2

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

$2 \cdot 3 x^{2 + 2} + 2 x^{2} x + 2 x^{2} \cdot - 1 - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.2.3

Add $2$ and $2$ .

$2 \cdot 3 x^{4} + 2 x^{2} x + 2 x^{2} \cdot - 1 - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.3

Multiply $2$ by $3$ .

$6 x^{4} + 2 x^{2} x + 2 x^{2} \cdot - 1 - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.4

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

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Step 2.1.4.1

Move $x$ .

$6 x^{4} + 2 (x \cdot x^{2}) + 2 x^{2} \cdot - 1 - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.4.2

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

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Step 2.1.4.2.1

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

$6 x^{4} + 2 (x^{1} x^{2}) + 2 x^{2} \cdot - 1 - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.4.2.2

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

$6 x^{4} + 2 x^{1 + 2} + 2 x^{2} \cdot - 1 - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.4.3

Add $1$ and $2$ .

$6 x^{4} + 2 x^{3} + 2 x^{2} \cdot - 1 - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.5

Multiply $- 1$ by $2$ .

$6 x^{4} + 2 x^{3} - 2 x^{2} - 7 x (3 x^{2}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$6 x^{4} + 2 x^{3} - 2 x^{2} - 7 \cdot 3 x \cdot x^{2} - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.7

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

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Step 2.1.7.1

Move $x^{2}$ .

$6 x^{4} + 2 x^{3} - 2 x^{2} - 7 \cdot 3 (x^{2} x) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.7.2

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

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Step 2.1.7.2.1

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

$6 x^{4} + 2 x^{3} - 2 x^{2} - 7 \cdot 3 (x^{2} x^{1}) - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.7.2.2

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

$6 x^{4} + 2 x^{3} - 2 x^{2} - 7 \cdot 3 x^{2 + 1} - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.7.3

Add $2$ and $1$ .

$6 x^{4} + 2 x^{3} - 2 x^{2} - 7 \cdot 3 x^{3} - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.8

Multiply $- 7$ by $3$ .

$6 x^{4} + 2 x^{3} - 2 x^{2} - 21 x^{3} - 7 x \cdot x - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.9

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

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Step 2.1.9.1

Move $x$ .

$6 x^{4} + 2 x^{3} - 2 x^{2} - 21 x^{3} - 7 (x \cdot x) - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.9.2

Multiply $x$ by $x$ .

$6 x^{4} + 2 x^{3} - 2 x^{2} - 21 x^{3} - 7 x^{2} - 7 x \cdot - 1 - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.10

Multiply $- 1$ by $- 7$ .

$6 x^{4} + 2 x^{3} - 2 x^{2} - 21 x^{3} - 7 x^{2} + 7 x - 1 (3 x^{2}) - 1 x - 1 \cdot - 1$

Step 2.1.11

Multiply $3$ by $- 1$ .

$6 x^{4} + 2 x^{3} - 2 x^{2} - 21 x^{3} - 7 x^{2} + 7 x - 3 x^{2} - 1 x - 1 \cdot - 1$

Step 2.1.12

Rewrite $- 1 x$ as $- x$ .

$6 x^{4} + 2 x^{3} - 2 x^{2} - 21 x^{3} - 7 x^{2} + 7 x - 3 x^{2} - x - 1 \cdot - 1$

Step 2.1.13

Multiply $- 1$ by $- 1$ .

$6 x^{4} + 2 x^{3} - 2 x^{2} - 21 x^{3} - 7 x^{2} + 7 x - 3 x^{2} - x + 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $21 x^{3}$ from $2 x^{3}$ .

$6 x^{4} - 19 x^{3} - 2 x^{2} - 7 x^{2} + 7 x - 3 x^{2} - x + 1$

Step 2.2.2

Subtract $7 x^{2}$ from $- 2 x^{2}$ .

$6 x^{4} - 19 x^{3} - 9 x^{2} + 7 x - 3 x^{2} - x + 1$

Step 2.2.3

Subtract $3 x^{2}$ from $- 9 x^{2}$ .

$6 x^{4} - 19 x^{3} - 12 x^{2} + 7 x - x + 1$

Step 2.2.4

Subtract $x$ from $7 x$ .

$6 x^{4} - 19 x^{3} - 12 x^{2} + 6 x + 1$