Precalculus Examples

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

Step 1

Expand $(x^{3} - 2 x + 1) (x^{3} + x^{2} - 5)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

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

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

Step 2.1.1.2

Add $3$ and $3$ .

$x^{6} + x^{3} x^{2} + x^{3} \cdot - 5 - 2 x \cdot x^{3} - 2 x \cdot x^{2} - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.2

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

Tap for more steps...

Step 2.1.2.1

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

$x^{6} + x^{3 + 2} + x^{3} \cdot - 5 - 2 x \cdot x^{3} - 2 x \cdot x^{2} - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.2.2

Add $3$ and $2$ .

$x^{6} + x^{5} + x^{3} \cdot - 5 - 2 x \cdot x^{3} - 2 x \cdot x^{2} - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.3

Move $- 5$ to the left of $x^{3}$ .

$x^{6} + x^{5} - 5 \cdot x^{3} - 2 x \cdot x^{3} - 2 x \cdot x^{2} - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.4

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

Tap for more steps...

Step 2.1.4.1

Move $x^{3}$ .

$x^{6} + x^{5} - 5 x^{3} - 2 (x^{3} x) - 2 x \cdot x^{2} - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.4.2

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

Tap for more steps...

Step 2.1.4.2.1

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

$x^{6} + x^{5} - 5 x^{3} - 2 (x^{3} x^{1}) - 2 x \cdot x^{2} - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.4.2.2

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

$x^{6} + x^{5} - 5 x^{3} - 2 x^{3 + 1} - 2 x \cdot x^{2} - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.4.3

Add $3$ and $1$ .

$x^{6} + x^{5} - 5 x^{3} - 2 x^{4} - 2 x \cdot x^{2} - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.5

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

Tap for more steps...

Step 2.1.5.1

Move $x^{2}$ .

$x^{6} + x^{5} - 5 x^{3} - 2 x^{4} - 2 (x^{2} x) - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.5.2

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

Tap for more steps...

Step 2.1.5.2.1

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

$x^{6} + x^{5} - 5 x^{3} - 2 x^{4} - 2 (x^{2} x^{1}) - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.5.2.2

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

$x^{6} + x^{5} - 5 x^{3} - 2 x^{4} - 2 x^{2 + 1} - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.5.3

Add $2$ and $1$ .

$x^{6} + x^{5} - 5 x^{3} - 2 x^{4} - 2 x^{3} - 2 x \cdot - 5 + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.6

Multiply $- 5$ by $- 2$ .

$x^{6} + x^{5} - 5 x^{3} - 2 x^{4} - 2 x^{3} + 10 x + 1 x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.7

Multiply $x^{3}$ by $1$ .

$x^{6} + x^{5} - 5 x^{3} - 2 x^{4} - 2 x^{3} + 10 x + x^{3} + 1 x^{2} + 1 \cdot - 5$

Step 2.1.8

Multiply $x^{2}$ by $1$ .

$x^{6} + x^{5} - 5 x^{3} - 2 x^{4} - 2 x^{3} + 10 x + x^{3} + x^{2} + 1 \cdot - 5$

Step 2.1.9

Multiply $- 5$ by $1$ .

$x^{6} + x^{5} - 5 x^{3} - 2 x^{4} - 2 x^{3} + 10 x + x^{3} + x^{2} - 5$

Step 2.2

Simplify by adding terms.

Tap for more steps...

Step 2.2.1

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

$x^{6} + x^{5} - 2 x^{4} - 7 x^{3} + 10 x + x^{3} + x^{2} - 5$

Step 2.2.2

Add $- 7 x^{3}$ and $x^{3}$ .

$x^{6} + x^{5} - 2 x^{4} - 6 x^{3} + 10 x + x^{2} - 5$

Step 2.2.3

Move $10 x$ .

$x^{6} + x^{5} - 2 x^{4} - 6 x^{3} + x^{2} + 10 x - 5$