Finite Math Examples

$f (x) = - 3 x - 5$ , $g (x) = - 3 x^{2} - 5 x + 4$ , $(f g) (x)$

Step 1

Evaluate $f g$ .

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Step 1.1

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

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

Step 1.2

Expand $(- 3 x - 5) (- 3 x^{2} - 5 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.3

Simplify terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x^{2}$ .

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

Step 1.3.1.2.2

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

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Step 1.3.1.2.2.1

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

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

Step 1.3.1.2.2.2

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

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

Step 1.3.1.2.3

Add $2$ and $1$ .

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

Step 1.3.1.3

Multiply $- 3$ by $- 3$ .

$9 x^{3} - 3 x (- 5 x) - 3 x \cdot 4 - 5 (- 3 x^{2}) - 5 (- 5 x) - 5 \cdot 4$

Step 1.3.1.4

Rewrite using the commutative property of multiplication.

$9 x^{3} - 3 \cdot - 5 x \cdot x - 3 x \cdot 4 - 5 (- 3 x^{2}) - 5 (- 5 x) - 5 \cdot 4$

Step 1.3.1.5

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

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Step 1.3.1.5.1

Move $x$ .

$9 x^{3} - 3 \cdot - 5 (x \cdot x) - 3 x \cdot 4 - 5 (- 3 x^{2}) - 5 (- 5 x) - 5 \cdot 4$

Step 1.3.1.5.2

Multiply $x$ by $x$ .

$9 x^{3} - 3 \cdot - 5 x^{2} - 3 x \cdot 4 - 5 (- 3 x^{2}) - 5 (- 5 x) - 5 \cdot 4$

Step 1.3.1.6

Multiply $- 3$ by $- 5$ .

$9 x^{3} + 15 x^{2} - 3 x \cdot 4 - 5 (- 3 x^{2}) - 5 (- 5 x) - 5 \cdot 4$

Step 1.3.1.7

Multiply $4$ by $- 3$ .

$9 x^{3} + 15 x^{2} - 12 x - 5 (- 3 x^{2}) - 5 (- 5 x) - 5 \cdot 4$

Step 1.3.1.8

Multiply $- 3$ by $- 5$ .

$9 x^{3} + 15 x^{2} - 12 x + 15 x^{2} - 5 (- 5 x) - 5 \cdot 4$

Step 1.3.1.9

Multiply $- 5$ by $- 5$ .

$9 x^{3} + 15 x^{2} - 12 x + 15 x^{2} + 25 x - 5 \cdot 4$

Step 1.3.1.10

Multiply $- 5$ by $4$ .

$9 x^{3} + 15 x^{2} - 12 x + 15 x^{2} + 25 x - 20$

Step 1.3.2

Simplify by adding terms.

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Step 1.3.2.1

Add $15 x^{2}$ and $15 x^{2}$ .

$9 x^{3} + 30 x^{2} - 12 x + 25 x - 20$

Step 1.3.2.2

Add $- 12 x$ and $25 x$ .

$9 x^{3} + 30 x^{2} + 13 x - 20$

Step 2

Evaluate $f g$ at $x$ .

$9 {(x)}^{3} + 30 {(x)}^{2} + 13 (x) - 20$

Step 3

Simplify $9 {(x)}^{3} + 30 {(x)}^{2} + 13 (x) - 20$ .

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Step 3.1

Remove parentheses.

$9 {(x)}^{3} + 30 {(x)}^{2} + 13 (x) - 20$

Step 3.2

Multiply $13$ by $x$ .

$9 x^{3} + 30 x^{2} + 13 x - 20$