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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 10 (page 765)

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.
$\frac{3 x^{4} + x^{2} - 2}{x^{3} + 8}$ .

Short Answer

Expert verified

The rational expression $\frac{3 x^{4} + x^{2} - 2}{x^{3} + 8}$ is improper.

The sum of a polynomial and a proper rational expression is $3 x + \frac{x^{2} - 24 x - 2}{x^{3} + 8}$ .

Step by step solution

01

Step 1. Given Information.

The rational expression is,

$\frac{3 x^{4} + x^{2} - 2}{x^{3} + 8}$ .

02

Step 2. Proper or Improper Rational Expression.

$\frac{3 x^{4} + x^{2} - 2}{x^{3} + 8}$ .

The degree of the numerator $= 4$ .

The degree of the denominator $= 3$ .

The degree of the numerator >The degree of the denominator.

It is an improper rational expression.

03

Step 3. Sum of polynomial and proper rational expression.

The improper rational expression is,

$\frac{3 x^{4} + x^{2} - 2}{x^{3} + 8}$ .

Add and subtract $24 x$ in the numerator of the expression,

$= \frac{3 x^{4} + 24 x - 24 x + x^{2} - 2}{x^{3} + 8} = \frac{3 x^{4} + 24 x}{x^{3} + 28} + \frac{x^{2} - 24 x - 2}{x^{3} + 8} = \frac{3 x (x^{3} + 8)}{x^{3} + 8} + \frac{x^{2} - 24 x - 2}{x^{3} + 8} = 3 x + \frac{x^{2} - 24 x - 2}{x^{3} + 8}$

The polynomial is $3 x$ and the proper rational expression is $\frac{x^{2} - 24 x - 2}{x^{3} + 8}$ .

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Most popular questions from this chapter

solve each system of equations. If the system has no solution, say that it is inconsistent.

x + y - z = 6

3x - 2y + z = -5

x + 3y - 2z = 14

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\{\begin{cases} 2 x + 4 y = \frac{2}{3} \\ 3 x - 5 y = - 1 \end{cases}$

The ratio of a to b is $\frac{2}{3}$ . The sum of a and b is $10$ . What is the ratio of $a + b$ to $b - a$ ?

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$y = \sqrt{x}; y = 2 - x$

Verify that the values of the variables listed are solutions of the system of equations.

$\{\begin{matrix} x - y = 3 \\ - 3 x + y = 1 \end{matrix} x = - 2, y = - 5; (- 2, - 5)$

See all solutions

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