Systems of Equations and Inequalities

Precalculus Enhanced with Graphing Utilities · 688 exercises

Q. 17

Write the partial fraction decomposition of each rational expression:

$\frac{x}{(x - 1) (x - 2)}$ .

3 step solution

Q. 18

Write the partial fraction decomposition of each rational expression.

$\frac{3 x}{(x + 2) (x - 4)}$ .

3 step solution

Q. 19

Write the partial fraction decomposition of each rational expression.

$\frac{x^{2}}{{(x - 1)}^{2} (x + 1)}$ .

4 step solution

Q. 20

Write the partial fraction decomposition of each rational expression.

$\frac{x + 1}{x^{2} (x - 2)}$ .

3 step solution

Q. 21

Write the partial fraction decomposition of each rational expression.

$\frac{1}{x^{3} - 8}$ .

4 step solution

Q.22

Write the partial fraction decomposition of each rational expression.

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

4 step solution

Q. 23

Write the partial fraction decomposition of each rational expression.

$\frac{x^{2}}{{(x - 1)}^{2} {(x + 1)}^{2}}$ .

4 step solution

Q. 24

Write the partial fraction decomposition of each rational expression.

$\frac{x + 1}{x^{2} {(x - 2)}^{2}}$ .

4 step solution

Q. 25

Write the partial fraction decomposition of each rational expression.

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

4 step solution

Q. 26

Write the partial fraction decomposition of each rational expression.

$\frac{x^{2} + x}{(x + 2) {(x - 1)}^{2}}$ .

4 step solution

Q. 27

Write the partial fraction decomposition of each rational expression.

$\frac{x + 4}{x^{2} (x^{2} + 4)}$

4 step solution

Q. 28

In Problems 13–46, write the partial fraction decomposition of each rational expression.

$\frac{10 x^{2} + 2 x}{{(x - 1)}^{2} (x^{2} + 2)}$

5 step solution

Q. 29

In Problems 13–46, write the partial fraction decomposition of each rational expression.

$\frac{x^{2} + 2 x + 3}{(x + 1) (x^{2} + 2 x + 4)}$

4 step solution

Q. 30

In Problems 13–46, write the partial fraction decomposition of each rational expression.

$\frac{x^{2} - 11 x - 18}{x (x^{2} + 3 x + 3)}$

4 step solution

Q. 31

In Problems 13–46, write the partial fraction decomposition of each rational expression.

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

4 step solution

Q. 32

In Problems 13–46, write the partial fraction decomposition of each rational expression.

$\frac{1}{(2 x + 3) (4 x - 1)}$

4 step solution

Q. 33

In Problems 13–46, write the partial fraction decomposition of each rational expression.

$\frac{x}{x^{2} + 2 x - 3}$

5 step solution

Q. 34

In Problems 13–46, write the partial fraction decomposition of each rational expression.

$\frac{x^{2} - x - 8}{(x + 1) (x^{2} + 5 x + 6)}$

4 step solution

Q. 35

In Problems 13–46, write the partial fraction decomposition of each rational expression.

$\frac{x^{2} + 2 x + 3}{{(x^{2} + 4)}^{2}}$

4 step solution

Q. 36

In Problems 13–46, write the partial fraction decomposition of each rational expression.

$\frac{x^{3} + 1}{{(x^{2} + 16)}^{2}}$

4 step solution

Q. 37

In Problems 13–46, write the partial fraction decomposition of each rational expression.

$\frac{7 x + 3}{x^{3} - 2 x^{2} - 3 x}$

4 step solution

Q. 38

Write the partial fraction decomposition of each rational expression.

3 step solution

Q. 39

Write the partial fraction decomposition of each rational expression.

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

3 step solution

Q. 40

Write the partial fraction decomposition of each rational expression.

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

3 step solution

Q. 41

Write the partial fraction decomposition of each rational expression.

$\frac{x^{3}}{{(x 2 + 16)}^{3}}$

2 step solution

Q. 42

Write the partial fraction decomposition of the rational expression.

$\frac{x^{2}}{{(x^{2} + 4)}^{3}}$

4 step solution

Q. 43

Write the partial fraction decomposition of the rational expression.

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

4 step solution

Q. 44

Write the partial fraction decomposition of the rational expression.

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

4 step solution

Q. 45

Write the partial fraction decomposition of the rational expression.

$\frac{2 x + 3}{x^{4} - 9 x^{2}}$

6 step solution

Q. 46

Write the partial fraction decomposition of the rational expression.

$\frac{x^{2} + 9}{x^{4} - 2 x^{2} - 8}$

6 step solution

Q. 47

Use the division algorithm to rewrite the improper fraction as the sum of a quotient and proper fraction. Find the partial fraction decomposition of the proper fraction. Finally, express the improper fraction as the sum of a quotient and the partial fraction decomposition.

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

4 step solution

Q. 48

Use the division algorithm to rewrite each improper fraction as the sum of a quotient and proper fraction. Find the partial fraction decomposition of the proper fraction. Finally, express the improper fraction as the sum of a quotient and the partial fraction decomposition.

$\frac{x^{3} - 3 x^{2} + 1}{x^{2} + 5 x + 6}$

3 step solution

Q. 48

In Problems 47–54, use the division algorithm to rewrite each improper fraction as the sum of a quotient and proper fraction. Find the partial fraction decomposition of the proper fraction. Finally, express the improper fraction as the sum of a quotient and the partial fraction decomposition

$\frac{x^{3} - 3 x^{2} + 1}{x^{2} + 5 x + 6}$

2 step solution

Q. 49

$\frac{x^{3}}{x^{2} + 1}$

2 step solution

Q. 50

$\frac{x^{3} + x}{x^{2} + 4}$

2 step solution

Q. 51

In Problems, use the division algorithm to rewrite each improper fraction as the sum of a quotient and proper fraction. Find the partial fraction decomposition of the proper fraction. Finally, express the improper fraction as the sum of a quotient and the partial fraction decomposition.

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

3 step solution

Q. 51

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

5 step solution

Q. 52

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

3 step solution

Q. 53

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

3 step solution

Q. 54

$\frac{x^{5} - x^{3} + x^{2} + 1}{x^{4} + 6 x^{2} + 9}$

3 step solution

Q. 1

Graph the equation: $y = 3 x + 2$

3 step solution

Q. 2

Graph the equation: $y + 4 = x^{2}$

3 step solution

Q. 3

Graph the equation: $y^{2} = x^{2} - 1$

3 step solution

Q. 4

Graph the equation: $x^{2} + 4 y^{2} = 4$

3 step solution

Q. 5

Graph each equation of the system. Then solve the system to find the points of intersection.

$\{\begin{cases} y = x^{2} + 1 \\ y = x + 1 \end{cases}$

6 step solution

Q. 5

True or False:

The lines $2 x + y = 4$ and $4 x + 2 y = 0$ are parallel.

2 step solution

Q. 6

Graph each equation of the system. Then solve the system to find the points of intersection.
$\{\begin{cases} y = x^{2} + 1 \\ y = 4 x + 1 \end{cases}$

6 step solution

Q. 6

The graph of $y = {(x - 2)}^{2}$ may be obtained by shifting the graph of ___ to the (left/right) ___ a distance of units.

2 step solution

Q. 7

Graph each equation of the system. Then solve the system to find the points of intersection.

$\{\begin{cases} y = \sqrt{36 - x^{2}} \\ y = 8 - x \end{cases}$

5 step solution

Q. 7

When graphing an inequality in two variables, if the inequality is strict use ___ ; if the inequality is nonstrict use a ___ mark.

2 step solution

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