Finite Math Examples

f (x) = \frac{{(3 x - 1)}^{3}}{{(x^{2} + 1)}^{2}}

$f (x) = \frac{{(3 x - 1)}^{3}}{{(x^{2} + 1)}^{2}}$

Step 1

A rational function is any function which can be written as the ratio of two polynomial functions where the denominator is not

0

$0$ .

f (x) = \frac{{(3 x - 1)}^{3}}{{(x^{2} + 1)}^{2}}

$f (x) = \frac{{(3 x - 1)}^{3}}{{(x^{2} + 1)}^{2}}$ is a rational function

Step 2

A rational function is proper when the degree of the numerator is less than the degree of the denominator, otherwise it is improper.

Degree of numerator is less than the degree of denominator implies a proper function

Degree of numerator is greater than the degree of denominator implies an improper function

Degree of numerator is equal to the degree of denominator implies an improper function

Step 3

Find the degree of the numerator.

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Step 3.1

Simplify and reorder the polynomial.

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Step 3.1.1

Use the Binomial Theorem.

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

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

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Apply the product rule to

3 x

$3 x$ .

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

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

Step 3.1.2.2

Raise

3

$3$ to the power of

3

$3$ .

27 x^{3} + 3 {(3 x)}^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} + 3 {(3 x)}^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}$

Step 3.1.2.3

Apply the product rule to

3 x

$3 x$ .

27 x^{3} + 3 (3^{2} x^{2}) \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} + 3 (3^{2} x^{2}) \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}$

Step 3.1.2.4

Multiply

3

$3$ by

3^{2}

$3^{2}$ by adding the exponents.

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Step 3.1.2.4.1

Move

3^{2}

$3^{2}$ .

27 x^{3} + 3^{2} \cdot 3 x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} + 3^{2} \cdot 3 x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}$

Step 3.1.2.4.2

Multiply

3^{2}

$3^{2}$ by

3

$3$ .

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Step 3.1.2.4.2.1

Raise

3

$3$ to the power of

1

$1$ .

27 x^{3} + 3^{2} \cdot 3^{1} x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} + 3^{2} \cdot 3^{1} x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}$

Step 3.1.2.4.2.2

Use the power rule

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

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

27 x^{3} + 3^{2 + 1} x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} + 3^{2 + 1} x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}$

27 x^{3} + 3^{2 + 1} x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} + 3^{2 + 1} x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}$

Step 3.1.2.4.3

Add

2

$2$ and

1

$1$ .

27 x^{3} + 3^{3} x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} + 3^{3} x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}$

27 x^{3} + 3^{3} x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} + 3^{3} x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}$

Step 3.1.2.5

Raise

3

$3$ to the power of

3

$3$ .

27 x^{3} + 27 x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} + 27 x^{2} \cdot - 1 + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}$

Step 3.1.2.6

Multiply

- 1

$- 1$ by

27

$27$ .

27 x^{3} - 27 x^{2} + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} - 27 x^{2} + 3 (3 x) {(- 1)}^{2} + {(- 1)}^{3}$

Step 3.1.2.7

Multiply

3

$3$ by

3

$3$ .

27 x^{3} - 27 x^{2} + 9 x {(- 1)}^{2} + {(- 1)}^{3}

$27 x^{3} - 27 x^{2} + 9 x {(- 1)}^{2} + {(- 1)}^{3}$

Step 3.1.2.8

Raise

- 1

$- 1$ to the power of

2

$2$ .

27 x^{3} - 27 x^{2} + 9 x \cdot 1 + {(- 1)}^{3}

$27 x^{3} - 27 x^{2} + 9 x \cdot 1 + {(- 1)}^{3}$

Step 3.1.2.9

Multiply

9

$9$ by

1

$1$ .

27 x^{3} - 27 x^{2} + 9 x + {(- 1)}^{3}

$27 x^{3} - 27 x^{2} + 9 x + {(- 1)}^{3}$

Step 3.1.2.10

Raise

- 1

$- 1$ to the power of

3

$3$ .

27 x^{3} - 27 x^{2} + 9 x - 1

$27 x^{3} - 27 x^{2} + 9 x - 1$

27 x^{3} - 27 x^{2} + 9 x - 1

$27 x^{3} - 27 x^{2} + 9 x - 1$

27 x^{3} - 27 x^{2} + 9 x - 1

$27 x^{3} - 27 x^{2} + 9 x - 1$

Step 3.2

The largest exponent is the degree of the polynomial.

3

$3$

3

$3$

Step 4

Find the degree of the denominator.

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Step 4.1

Simplify and reorder the polynomial.

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Step 4.1.1

Rewrite

{(x^{2} + 1)}^{2}

${(x^{2} + 1)}^{2}$ as

(x^{2} + 1) (x^{2} + 1)

$(x^{2} + 1) (x^{2} + 1)$ .

$(x^{2} + 1) (x^{2} + 1)$

Step 4.1.2

Expand

$(x^{2} + 1) (x^{2} + 1)$ using the FOIL Method.

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Step 4.1.2.1

Apply the distributive property.

$x^{2} (x^{2} + 1) + 1 (x^{2} + 1)$

Step 4.1.2.2

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 1 + 1 (x^{2} + 1)$

Step 4.1.2.3

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1$

Step 4.1.3

Simplify and combine like terms.

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Step 4.1.3.1

Simplify each term.

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Step 4.1.3.1.1

Multiply

$x^{2}$ by

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

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Step 4.1.3.1.1.1

Use the power rule

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

$x^{2 + 2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1$

Step 4.1.3.1.1.2

Add

$2$ and

$2$ .

$x^{4} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1$

Step 4.1.3.1.2

Multiply

$x^{2}$ by

$1$ .

$x^{4} + x^{2} + 1 x^{2} + 1 \cdot 1$

Step 4.1.3.1.3

Multiply

$x^{2}$ by

$1$ .

$x^{4} + x^{2} + x^{2} + 1 \cdot 1$

Step 4.1.3.1.4

Multiply

$1$ by

$1$ .

$x^{4} + x^{2} + x^{2} + 1$

Step 4.1.3.2

Add

$x^{2}$ and

$x^{2}$ .

$x^{4} + 2 x^{2} + 1$

Step 4.2

The largest exponent is the degree of the polynomial.

$4$

Step 5

The degree of the numerator

$3$ is less than the degree of the denominator

$4$ .

$3 < 4$

Step 6

The degree of the numerator is less than the degree of the denominator, which means that

$f (x)$ is a proper function.

Proper