Precalculus Examples

Given that

f (x) = x^{2} - 13 x + 30

$f (x) = x^{2} - 13 x + 30$ and

g (x) = x - 3

$g (x) = x - 3$ ; find

f (x) \cdot g (x)

$f (x) \cdot g (x)$ and express the result in standard form.

Step 1

Replace the function designators in

f (x) \cdot (g (x))

$f (x) \cdot (g (x))$ with the actual functions.

(x^{2} - 13 x + 30) \cdot (x - 3)

$(x^{2} - 13 x + 30) \cdot (x - 3)$

Step 2

Simplify

(x^{2} - 13 x + 30) \cdot (x - 3)

$(x^{2} - 13 x + 30) \cdot (x - 3)$ .

Tap for more steps...

Step 2.1

Expand

(x^{2} - 13 x + 30) (x - 3)

$(x^{2} - 13 x + 30) (x - 3)$ by multiplying each term in the first expression by each term in the second expression.

x^{2} x + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3

$x^{2} x + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3$

Step 2.2

Simplify terms.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply

x^{2}

$x^{2}$ by

x

$x$ by adding the exponents.

Tap for more steps...

Step 2.2.1.1.1

Multiply

x^{2}

$x^{2}$ by

x

$x$ .

Tap for more steps...

Step 2.2.1.1.1.1

Raise

x

$x$ to the power of

1

$1$ .

x^{2} x^{1} + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3

$x^{2} x^{1} + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3$

Step 2.2.1.1.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

x^{2 + 1} + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3

$x^{2 + 1} + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3$

x^{2 + 1} + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3

$x^{2 + 1} + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3$

Step 2.2.1.1.2

Add

2

$2$ and

1

$1$ .

x^{3} + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3

$x^{3} + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3$

x^{3} + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3

$x^{3} + x^{2} \cdot - 3 - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3$

Step 2.2.1.2

Move

- 3

$- 3$ to the left of

x^{2}

$x^{2}$ .

x^{3} - 3 \cdot x^{2} - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3

$x^{3} - 3 \cdot x^{2} - 13 x \cdot x - 13 x \cdot - 3 + 30 x + 30 \cdot - 3$

Step 2.2.1.3

Multiply

x

$x$ by

x

$x$ by adding the exponents.

Tap for more steps...

Step 2.2.1.3.1

Move

x

$x$ .

x^{3} - 3 x^{2} - 13 (x \cdot x) - 13 x \cdot - 3 + 30 x + 30 \cdot - 3

$x^{3} - 3 x^{2} - 13 (x \cdot x) - 13 x \cdot - 3 + 30 x + 30 \cdot - 3$

Step 2.2.1.3.2

Multiply

x

$x$ by

x

$x$ .

x^{3} - 3 x^{2} - 13 x^{2} - 13 x \cdot - 3 + 30 x + 30 \cdot - 3

$x^{3} - 3 x^{2} - 13 x^{2} - 13 x \cdot - 3 + 30 x + 30 \cdot - 3$

x^{3} - 3 x^{2} - 13 x^{2} - 13 x \cdot - 3 + 30 x + 30 \cdot - 3

$x^{3} - 3 x^{2} - 13 x^{2} - 13 x \cdot - 3 + 30 x + 30 \cdot - 3$

Step 2.2.1.4

Multiply

- 3

$- 3$ by

- 13

$- 13$ .

x^{3} - 3 x^{2} - 13 x^{2} + 39 x + 30 x + 30 \cdot - 3

$x^{3} - 3 x^{2} - 13 x^{2} + 39 x + 30 x + 30 \cdot - 3$

Step 2.2.1.5

Multiply

30

$30$ by

- 3

$- 3$ .

x^{3} - 3 x^{2} - 13 x^{2} + 39 x + 30 x - 90

$x^{3} - 3 x^{2} - 13 x^{2} + 39 x + 30 x - 90$

x^{3} - 3 x^{2} - 13 x^{2} + 39 x + 30 x - 90

$x^{3} - 3 x^{2} - 13 x^{2} + 39 x + 30 x - 90$

Step 2.2.2

Simplify by adding terms.

Tap for more steps...

Step 2.2.2.1

Subtract

13 x^{2}

$13 x^{2}$ from

- 3 x^{2}

$- 3 x^{2}$ .

x^{3} - 16 x^{2} + 39 x + 30 x - 90

$x^{3} - 16 x^{2} + 39 x + 30 x - 90$

Step 2.2.2.2

Add

39 x

$39 x$ and

30 x

$30 x$ .

x^{3} - 16 x^{2} + 69 x - 90

$x^{3} - 16 x^{2} + 69 x - 90$

x^{3} - 16 x^{2} + 69 x - 90

$x^{3} - 16 x^{2} + 69 x - 90$

x^{3} - 16 x^{2} + 69 x - 90

$x^{3} - 16 x^{2} + 69 x - 90$

x^{3} - 16 x^{2} + 69 x - 90

$x^{3} - 16 x^{2} + 69 x - 90$