Determine if Proper or Improper

$g (x) = \frac{12 x^{3} + 7 x^{2} - 1}{x^{2} + x}$

Step 1

A rational function is any function which can be written as the ratio of two polynomial functions where the denominator is not $0$ .

$g (x) = \frac{12 x^{3} + 7 x^{2} - 1}{x^{2} + x}$ 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

Remove parentheses.

$12 x^{3} + 7 x^{2} - 1$

Step 3.2

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$12 x^{3} \to 3$

$7 x^{2} \to 2$

$- 1 \to 0$

Step 3.3

The largest exponent is the degree of the polynomial.

$3$

Step 4

Find the degree of the denominator.

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Step 4.1

Remove parentheses.

$x^{2} + x$

Step 4.2

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$x^{2} \to 2$

$x \to 1$

Step 4.3

The largest exponent is the degree of the polynomial.

$2$

Step 5

The degree of the numerator $3$ is greater than the degree of the denominator $2$ .

$3 > 2$

Step 6

The degree of the numerator is greater than the degree of the denominator, which means that $g (x)$ is an improper function.

Improper

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