Algebra Examples

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

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

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Move $x^{2}$ .

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

Step 3.1.1.2

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

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

Step 3.1.1.3

Add $2$ and $2$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

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

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Step 3.1.3.1

Move $x$ .

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

Step 3.1.3.2

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

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Step 3.1.3.2.1

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

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

Step 3.1.3.2.2

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

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

Step 3.1.3.3

Add $1$ and $2$ .

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

Step 3.1.4

Multiply $3$ by $- 1$ .

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

Step 3.1.5

Multiply $- 1$ by $3$ .

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

Step 3.1.6

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

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Step 3.1.6.1

Move $x^{2}$ .

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

Step 3.1.6.2

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

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Step 3.1.6.2.1

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

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

Step 3.1.6.2.2

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

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

Step 3.1.6.3

Add $2$ and $1$ .

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

Step 3.1.7

Rewrite using the commutative property of multiplication.

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

Step 3.1.8

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

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Step 3.1.8.1

Move $x$ .

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

Step 3.1.8.2

Multiply $x$ by $x$ .

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

Step 3.1.9

Multiply $2$ by $- 1$ .

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

Step 3.1.10

Multiply $- 1$ by $2$ .

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

Step 3.1.11

Multiply $- 1$ by $- 3$ .

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

Step 3.1.12

Multiply $- 3$ by $- 1$ .

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

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Add $- 3 x^{3}$ and $2 x^{3}$ .

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

Step 3.2.2

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

$3 x^{4} - x^{3} - 5 x^{2} - 2 x - 3 x^{2} + 3 x + 3$

Step 3.2.3

Subtract $3 x^{2}$ from $- 5 x^{2}$ .

$3 x^{4} - x^{3} - 8 x^{2} - 2 x + 3 x + 3$

Step 3.2.4

Add $- 2 x$ and $3 x$ .

$3 x^{4} - x^{3} - 8 x^{2} + x + 3$