Algebra Examples

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

Step 1

Expand $(x^{2} + x + 3) (x^{3} - x^{2} + 4)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 2.1.1.1

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

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

Step 2.1.1.2

Add $2$ and $3$ .

$x^{5} + x^{2} (- x^{2}) + x^{2} \cdot 4 + x \cdot x^{3} + x (- x^{2}) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$x^{5} - x^{2} x^{2} + x^{2} \cdot 4 + x \cdot x^{3} + x (- x^{2}) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.3

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

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Step 2.1.3.1

Move $x^{2}$ .

$x^{5} - (x^{2} x^{2}) + x^{2} \cdot 4 + x \cdot x^{3} + x (- x^{2}) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.3.2

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

$x^{5} - x^{2 + 2} + x^{2} \cdot 4 + x \cdot x^{3} + x (- x^{2}) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.3.3

Add $2$ and $2$ .

$x^{5} - x^{4} + x^{2} \cdot 4 + x \cdot x^{3} + x (- x^{2}) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.4

Move $4$ to the left of $x^{2}$ .

$x^{5} - x^{4} + 4 \cdot x^{2} + x \cdot x^{3} + x (- x^{2}) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.5

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

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Step 2.1.5.1

Multiply $x$ by $x^{3}$ .

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Step 2.1.5.1.1

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

$x^{5} - x^{4} + 4 x^{2} + x^{1} x^{3} + x (- x^{2}) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.5.1.2

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

$x^{5} - x^{4} + 4 x^{2} + x^{1 + 3} + x (- x^{2}) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.5.2

Add $1$ and $3$ .

$x^{5} - x^{4} + 4 x^{2} + x^{4} + x (- x^{2}) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$x^{5} - x^{4} + 4 x^{2} + x^{4} - x \cdot x^{2} + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.7

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

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Step 2.1.7.1

Move $x^{2}$ .

$x^{5} - x^{4} + 4 x^{2} + x^{4} - (x^{2} x) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

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 $1$ .

$x^{5} - x^{4} + 4 x^{2} + x^{4} - (x^{2} x^{1}) + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.7.2.2

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

$x^{5} - x^{4} + 4 x^{2} + x^{4} - x^{2 + 1} + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.7.3

Add $2$ and $1$ .

$x^{5} - x^{4} + 4 x^{2} + x^{4} - x^{3} + x \cdot 4 + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.8

Move $4$ to the left of $x$ .

$x^{5} - x^{4} + 4 x^{2} + x^{4} - x^{3} + 4 \cdot x + 3 x^{3} + 3 (- x^{2}) + 3 \cdot 4$

Step 2.1.9

Multiply $- 1$ by $3$ .

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

Step 2.1.10

Multiply $3$ by $4$ .

$x^{5} - x^{4} + 4 x^{2} + x^{4} - x^{3} + 4 x + 3 x^{3} - 3 x^{2} + 12$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Combine the opposite terms in $x^{5} - x^{4} + 4 x^{2} + x^{4} - x^{3} + 4 x + 3 x^{3} - 3 x^{2} + 12$ .

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Step 2.2.1.1

Add $- x^{4}$ and $x^{4}$ .

$x^{5} + 4 x^{2} + 0 - x^{3} + 4 x + 3 x^{3} - 3 x^{2} + 12$

Step 2.2.1.2

Add $x^{5} + 4 x^{2}$ and $0$ .

$x^{5} + 4 x^{2} - x^{3} + 4 x + 3 x^{3} - 3 x^{2} + 12$

Step 2.2.2

Subtract $3 x^{2}$ from $4 x^{2}$ .

$x^{5} - x^{3} + 4 x + 3 x^{3} + x^{2} + 12$

Step 2.2.3

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

$x^{5} + 2 x^{3} + 4 x + x^{2} + 12$

Step 2.2.4

Move $4 x$ .

$x^{5} + 2 x^{3} + x^{2} + 4 x + 12$