Algebra Examples

$h (n) = - 3 n^{3} + 2 n^{2}$ $g (n) = n + 1$

Step 1

Set up the composite result function.

$h (g (n))$

Step 2

Evaluate $h (n + 1)$ by substituting in the value of $g$ into $h$ .

$h (n + 1) = - 3 {(n + 1)}^{3} + 2 {(n + 1)}^{2}$

Step 3

Simplify each term.

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Step 3.1

Use the Binomial Theorem.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $3$ by $1$ .

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

Step 3.2.2

One to any power is one.

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

Step 3.2.3

Multiply $3$ by $1$ .

$h (n + 1) = - 3 (n^{3} + 3 n^{2} + 3 n + 1^{3}) + 2 {(n + 1)}^{2}$

Step 3.2.4

One to any power is one.

$h (n + 1) = - 3 (n^{3} + 3 n^{2} + 3 n + 1) + 2 {(n + 1)}^{2}$

Step 3.3

Apply the distributive property.

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

Step 3.4

Simplify.

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Step 3.4.1

Multiply $3$ by $- 3$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 3 (3 n) - 3 \cdot 1 + 2 {(n + 1)}^{2}$

Step 3.4.2

Multiply $3$ by $- 3$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 \cdot 1 + 2 {(n + 1)}^{2}$

Step 3.4.3

Multiply $- 3$ by $1$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 {(n + 1)}^{2}$

Step 3.5

Rewrite ${(n + 1)}^{2}$ as $(n + 1) (n + 1)$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 ((n + 1) (n + 1))$

Step 3.6

Expand $(n + 1) (n + 1)$ using the FOIL Method.

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Step 3.6.1

Apply the distributive property.

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 (n (n + 1) + 1 (n + 1))$

Step 3.6.2

Apply the distributive property.

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 (n \cdot n + n \cdot 1 + 1 (n + 1))$

Step 3.6.3

Apply the distributive property.

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 (n \cdot n + n \cdot 1 + 1 n + 1 \cdot 1)$

Step 3.7

Simplify and combine like terms.

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Step 3.7.1

Simplify each term.

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Step 3.7.1.1

Multiply $n$ by $n$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 (n^{2} + n \cdot 1 + 1 n + 1 \cdot 1)$

Step 3.7.1.2

Multiply $n$ by $1$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 (n^{2} + n + 1 n + 1 \cdot 1)$

Step 3.7.1.3

Multiply $n$ by $1$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 (n^{2} + n + n + 1 \cdot 1)$

Step 3.7.1.4

Multiply $1$ by $1$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 (n^{2} + n + n + 1)$

Step 3.7.2

Add $n$ and $n$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 (n^{2} + 2 n + 1)$

Step 3.8

Apply the distributive property.

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 n^{2} + 2 (2 n) + 2 \cdot 1$

Step 3.9

Simplify.

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Step 3.9.1

Multiply $2$ by $2$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 n^{2} + 4 n + 2 \cdot 1$

Step 3.9.2

Multiply $2$ by $1$ .

$h (n + 1) = - 3 n^{3} - 9 n^{2} - 9 n - 3 + 2 n^{2} + 4 n + 2$

Step 4

Simplify by adding terms.

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Step 4.1

Add $- 9 n^{2}$ and $2 n^{2}$ .

$h (n + 1) = - 3 n^{3} - 7 n^{2} - 9 n - 3 + 4 n + 2$

Step 4.2

Add $- 9 n$ and $4 n$ .

$h (n + 1) = - 3 n^{3} - 7 n^{2} - 5 n - 3 + 2$

Step 4.3

Add $- 3$ and $2$ .

$h (n + 1) = - 3 n^{3} - 7 n^{2} - 5 n - 1$