Precalculus Examples

Given that $f (x) = x^{2} + 4 x - 60$ and $g (x) = x - 6$ ; find $(f \cdot g) (x)$ and express the result in standard form.

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.

$(x^{2} + 4 x - 60) \cdot (x - 6)$

Step 1.2

Expand $(x^{2} + 4 x - 60) (x - 6)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x + x^{2} \cdot - 6 + 4 x \cdot x + 4 x \cdot - 6 - 60 x - 60 \cdot - 6$

Step 1.3

Simplify terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

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

Tap for more steps...

Step 1.3.1.1.1

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

Tap for more steps...

Step 1.3.1.1.1.1

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

$x^{2} x^{1} + x^{2} \cdot - 6 + 4 x \cdot x + 4 x \cdot - 6 - 60 x - 60 \cdot - 6$

Step 1.3.1.1.1.2

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

$x^{2 + 1} + x^{2} \cdot - 6 + 4 x \cdot x + 4 x \cdot - 6 - 60 x - 60 \cdot - 6$

Step 1.3.1.1.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot - 6 + 4 x \cdot x + 4 x \cdot - 6 - 60 x - 60 \cdot - 6$

Step 1.3.1.2

Move $- 6$ to the left of $x^{2}$ .

$x^{3} - 6 \cdot x^{2} + 4 x \cdot x + 4 x \cdot - 6 - 60 x - 60 \cdot - 6$

Step 1.3.1.3

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

Tap for more steps...

Step 1.3.1.3.1

Move $x$ .

$x^{3} - 6 x^{2} + 4 (x \cdot x) + 4 x \cdot - 6 - 60 x - 60 \cdot - 6$

Step 1.3.1.3.2

Multiply $x$ by $x$ .

$x^{3} - 6 x^{2} + 4 x^{2} + 4 x \cdot - 6 - 60 x - 60 \cdot - 6$

Step 1.3.1.4

Multiply $- 6$ by $4$ .

$x^{3} - 6 x^{2} + 4 x^{2} - 24 x - 60 x - 60 \cdot - 6$

Step 1.3.1.5

Multiply $- 60$ by $- 6$ .

$x^{3} - 6 x^{2} + 4 x^{2} - 24 x - 60 x + 360$

Step 1.3.2

Simplify by adding terms.

Tap for more steps...

Step 1.3.2.1

Add $- 6 x^{2}$ and $4 x^{2}$ .

$x^{3} - 2 x^{2} - 24 x - 60 x + 360$

Step 1.3.2.2

Subtract $60 x$ from $- 24 x$ .

$x^{3} - 2 x^{2} - 84 x + 360$

Step 2

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

${(x)}^{3} - 2 {(x)}^{2} - 84 x + 360$

Step 3

Simplify ${(x)}^{3} - 2 {(x)}^{2} - 84 x + 360$ .

Tap for more steps...

Step 3.1

Remove parentheses.

${(x)}^{3} - 2 {(x)}^{2} - 84 x + 360$

Step 3.2

Remove parentheses.

$x^{3} - 2 x^{2} - 84 x + 360$