Precalculus Examples

$g (n) = n^{2} + 4 + 2 n$ $h (n) = - 3 n + 2$ Find $(g \cdot h) (1)$

Step 1

Evaluate $g \cdot h$ .

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Step 1.1

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

$(n^{2} + 4 + 2 n) \cdot (- 3 n + 2)$

Step 1.2

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

$n^{2} (- 3 n) + n^{2} \cdot 2 + 4 (- 3 n) + 4 \cdot 2 + 2 n (- 3 n) + 2 n \cdot 2$

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.

$- 3 n^{2} n + n^{2} \cdot 2 + 4 (- 3 n) + 4 \cdot 2 + 2 n (- 3 n) + 2 n \cdot 2$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $n$ .

$- 3 (n \cdot n^{2}) + n^{2} \cdot 2 + 4 (- 3 n) + 4 \cdot 2 + 2 n (- 3 n) + 2 n \cdot 2$

Step 1.3.1.2.2

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

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Step 1.3.1.2.2.1

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

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

Step 1.3.1.2.2.2

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

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

Step 1.3.1.2.3

Add $1$ and $2$ .

$- 3 n^{3} + n^{2} \cdot 2 + 4 (- 3 n) + 4 \cdot 2 + 2 n (- 3 n) + 2 n \cdot 2$

Step 1.3.1.3

Move $2$ to the left of $n^{2}$ .

$- 3 n^{3} + 2 \cdot n^{2} + 4 (- 3 n) + 4 \cdot 2 + 2 n (- 3 n) + 2 n \cdot 2$

Step 1.3.1.4

Multiply $- 3$ by $4$ .

$- 3 n^{3} + 2 n^{2} - 12 n + 4 \cdot 2 + 2 n (- 3 n) + 2 n \cdot 2$

Step 1.3.1.5

Multiply $4$ by $2$ .

$- 3 n^{3} + 2 n^{2} - 12 n + 8 + 2 n (- 3 n) + 2 n \cdot 2$

Step 1.3.1.6

Rewrite using the commutative property of multiplication.

$- 3 n^{3} + 2 n^{2} - 12 n + 8 + 2 \cdot - 3 n \cdot n + 2 n \cdot 2$

Step 1.3.1.7

Multiply $n$ by $n$ by adding the exponents.

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Step 1.3.1.7.1

Move $n$ .

$- 3 n^{3} + 2 n^{2} - 12 n + 8 + 2 \cdot - 3 (n \cdot n) + 2 n \cdot 2$

Step 1.3.1.7.2

Multiply $n$ by $n$ .

$- 3 n^{3} + 2 n^{2} - 12 n + 8 + 2 \cdot - 3 n^{2} + 2 n \cdot 2$

Step 1.3.1.8

Multiply $2$ by $- 3$ .

$- 3 n^{3} + 2 n^{2} - 12 n + 8 - 6 n^{2} + 2 n \cdot 2$

Step 1.3.1.9

Multiply $2$ by $2$ .

$- 3 n^{3} + 2 n^{2} - 12 n + 8 - 6 n^{2} + 4 n$

Step 1.3.2

Simplify by adding terms.

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Step 1.3.2.1

Subtract $6 n^{2}$ from $2 n^{2}$ .

$- 3 n^{3} - 4 n^{2} - 12 n + 8 + 4 n$

Step 1.3.2.2

Add $- 12 n$ and $4 n$ .

$- 3 n^{3} - 4 n^{2} - 8 n + 8$

Step 2

Evaluate $g \cdot h$ at $1$ .

$- 3 {(1)}^{3} - 4 {(1)}^{2} - 8 \cdot 1 + 8$

Step 3

Simplify $- 3 {(1)}^{3} - 4 {(1)}^{2} - 8 \cdot 1 + 8$ .

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Step 3.1

Remove parentheses.

$- 3 {(1)}^{3} - 4 {(1)}^{2} - 8 \cdot 1 + 8$

Step 3.2

Simplify each term.

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Step 3.2.1

One to any power is one.

$- 3 \cdot 1 - 4 {(1)}^{2} - 8 \cdot 1 + 8$

Step 3.2.2

Multiply $- 3$ by $1$ .

$- 3 - 4 {(1)}^{2} - 8 \cdot 1 + 8$

Step 3.2.3

One to any power is one.

$- 3 - 4 \cdot 1 - 8 \cdot 1 + 8$

Step 3.2.4

Multiply $- 4$ by $1$ .

$- 3 - 4 - 8 \cdot 1 + 8$

Step 3.2.5

Multiply $- 8$ by $1$ .

$- 3 - 4 - 8 + 8$

Step 3.3

Simplify by adding and subtracting.

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Step 3.3.1

Subtract $4$ from $- 3$ .

$- 7 - 8 + 8$

Step 3.3.2

Subtract $8$ from $- 7$ .

$- 15 + 8$

Step 3.3.3

Add $- 15$ and $8$ .

$- 7$