Q. 9

Question

Tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.

$\frac{5 x^{3} + 2 x - 1}{x^{2} - 4}$ .

Step-by-Step Solution

Verified

Answer

The rational expression $\frac{5 x^{3} + 2 x - 1}{x^{2} - 4}$ is improper.

The sum of a polynomial and a proper rational expression is $5 x + \frac{22 x - 1}{x^{2} - 4}$ .

1Step 1. Given Information.

The rational expression is,

$\frac{5 x^{3} + 2 x - 1}{x^{2} - 4}$ .

2Step 2. Improper and Proper Rational Expression.

$\frac{5 x^{3} + 2 x - 1}{x^{2} - 4}$ .

The degree of the numerator $= 3$ .

The degree of the denominator $= 2$

Degree of the numerator>Degree of the denominator.

It is an improper rational expression.

3Step 3. Sum of Polynomial and Proper Rational Expression.

The improper rational expression is,

$\frac{5 x^{3} + 2 x - 1}{x^{2} - 4}$ .

Dividing the expression, we get,

$= 5 x + \frac{22 x - 1}{x^{2} - 4}$ .

The polynomial is $5 x$ and the proper rational expression is $\frac{22 x - 1}{x^{2} - 4}$ .

Q. 8

Q. 10

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