Finite Math Examples

$P (x) = x^{4} - 12 x^{2} - 67$ , $d (x) = x + 4$

Step 1

Set up the composite result function.

$P (d (x))$

Step 2

Evaluate $P (x + 4)$ by substituting in the value of $d$ into $P$ .

$P (x + 4) = {(x + 4)}^{4} - 12 {(x + 4)}^{2} - 67$

Step 3

Simplify each term.

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Step 3.1

Use the Binomial Theorem.

$P (x + 4) = x^{4} + 4 x^{3} \cdot 4 + 6 x^{2} \cdot 4^{2} + 4 x \cdot 4^{3} + 4^{4} - 12 {(x + 4)}^{2} - 67$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $4$ by $4$ .

$P (x + 4) = x^{4} + 16 x^{3} + 6 x^{2} \cdot 4^{2} + 4 x \cdot 4^{3} + 4^{4} - 12 {(x + 4)}^{2} - 67$

Step 3.2.2

Raise $4$ to the power of $2$ .

$P (x + 4) = x^{4} + 16 x^{3} + 6 x^{2} \cdot 16 + 4 x \cdot 4^{3} + 4^{4} - 12 {(x + 4)}^{2} - 67$

Step 3.2.3

Multiply $16$ by $6$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 4 x \cdot 4^{3} + 4^{4} - 12 {(x + 4)}^{2} - 67$

Step 3.2.4

Multiply $4$ by $4^{3}$ by adding the exponents.

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Step 3.2.4.1

Move $4^{3}$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 4^{3} \cdot (4 x) + 4^{4} - 12 {(x + 4)}^{2} - 67$

Step 3.2.4.2

Multiply $4^{3}$ by $4$ .

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Step 3.2.4.2.1

Raise $4$ to the power of $1$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 4^{3} \cdot (4 x) + 4^{4} - 12 {(x + 4)}^{2} - 67$

Step 3.2.4.2.2

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

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 4^{3 + 1} x + 4^{4} - 12 {(x + 4)}^{2} - 67$

Step 3.2.4.3

Add $3$ and $1$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 4^{4} x + 4^{4} - 12 {(x + 4)}^{2} - 67$

Step 3.2.5

Raise $4$ to the power of $4$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 4^{4} - 12 {(x + 4)}^{2} - 67$

Step 3.2.6

Raise $4$ to the power of $4$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 {(x + 4)}^{2} - 67$

Step 3.3

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 ((x + 4) (x + 4)) - 67$

Step 3.4

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 3.4.1

Apply the distributive property.

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 (x (x + 4) + 4 (x + 4)) - 67$

Step 3.4.2

Apply the distributive property.

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 (x \cdot x + x \cdot 4 + 4 (x + 4)) - 67$

Step 3.4.3

Apply the distributive property.

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) - 67$

Step 3.5

Simplify and combine like terms.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Multiply $x$ by $x$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) - 67$

Step 3.5.1.2

Move $4$ to the left of $x$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) - 67$

Step 3.5.1.3

Multiply $4$ by $4$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 (x^{2} + 4 x + 4 x + 16) - 67$

Step 3.5.2

Add $4 x$ and $4 x$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 (x^{2} + 8 x + 16) - 67$

Step 3.6

Apply the distributive property.

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 x^{2} - 12 (8 x) - 12 \cdot 16 - 67$

Step 3.7

Simplify.

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Step 3.7.1

Multiply $8$ by $- 12$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 x^{2} - 96 x - 12 \cdot 16 - 67$

Step 3.7.2

Multiply $- 12$ by $16$ .

$P (x + 4) = x^{4} + 16 x^{3} + 96 x^{2} + 256 x + 256 - 12 x^{2} - 96 x - 192 - 67$

Step 4

Simplify by adding terms.

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Step 4.1

Subtract $12 x^{2}$ from $96 x^{2}$ .

$P (x + 4) = x^{4} + 16 x^{3} + 84 x^{2} + 256 x + 256 - 96 x - 192 - 67$

Step 4.2

Subtract $96 x$ from $256 x$ .

$P (x + 4) = x^{4} + 16 x^{3} + 84 x^{2} + 160 x + 256 - 192 - 67$

Step 4.3

Simplify by subtracting numbers.

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Step 4.3.1

Subtract $192$ from $256$ .

$P (x + 4) = x^{4} + 16 x^{3} + 84 x^{2} + 160 x + 64 - 67$

Step 4.3.2

Subtract $67$ from $64$ .

$P (x + 4) = x^{4} + 16 x^{3} + 84 x^{2} + 160 x - 3$