Precalculus Examples

$f (x) = 2 x^{3} - 5 x^{2}$ $g (x) = 2 x - 1$ Find $(f \cdot g) (x)$

Step 1

Evaluate $f \cdot g$ .

Tap for more steps...

Step 1.1

Replace the function designators in $f \cdot g$ with the actual functions.

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

Step 1.2

Expand $(2 x^{3} - 5 x^{2}) (2 x - 1)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

Tap for more steps...

Step 1.3.1.2.1

Move $x$ .

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

Step 1.3.1.2.2

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

Tap for more steps...

Step 1.3.1.2.2.1

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

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

Step 1.3.1.2.2.2

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

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

Step 1.3.1.2.3

Add $1$ and $3$ .

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

Step 1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.3.1.4

Multiply $- 1$ by $2$ .

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

Step 1.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.6

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

Tap for more steps...

Step 1.3.1.6.1

Move $x$ .

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

Step 1.3.1.6.2

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

Tap for more steps...

Step 1.3.1.6.2.1

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

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

Step 1.3.1.6.2.2

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

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

Step 1.3.1.6.3

Add $1$ and $2$ .

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

Step 1.3.1.7

Multiply $- 5$ by $2$ .

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

Step 1.3.1.8

Multiply $- 1$ by $- 5$ .

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

Step 1.3.2

Subtract $10 x^{3}$ from $- 2 x^{3}$ .

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

Step 2

Evaluate $f \cdot g$ at $x$ .

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

Step 3

Simplify $4 {(x)}^{4} - 12 {(x)}^{3} + 5 {(x)}^{2}$ .

Tap for more steps...

Step 3.1

Remove parentheses.

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

Step 3.2

Remove parentheses.

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