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# 4.E: Polynomial and Rational Functions (Exercises)

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Exercise $$\PageIndex{1}$$

Evaluate

1. Given $$f ( x ) = 2 x ^ { 2 } - x + 6$$, find $$f ( - 3 ) , f ( 0 )$$, and $$f ( 10 )$$.
2. Given $$g ( x ) = - x ^ { 2 } + 4 x - 1$$, find $$g ( - 1 ) , g ( 0 )$$, and $$g(3)$$.
3. Given $$h ( t ) = - t ^ { 3 } - 2 t ^ { 2 } + 3$$, find $$h ( - 3 ) , h ( 0 )$$, and $$h(2)$$.
4. Given $$p ( x ) = x ^ { 4 } - 2 x ^ { 2 } + x$$, find $$p ( - 1 ) , p ( 0 )$$, and $$p(2)$$.
5. The following graph gives the height $$h (t)$$ in feet of a projectile over time $$t$$ in seconds

(a) Use the graph to determine the height of the projectile at $$2.5$$ seconds.

(b) At what time does the projectile reach its maximum height?

(c) How long does it take the projectile to return to the ground?

6. Given the graph of the function $$f$$, find $$f ( - 9 ) , f ( - 3 )$$, and $$f(12)$$.

7. From the ground, a bullet is fired straight up into the air at $$340$$ meters per second. Ignoring the effects of air friction, write a function that models the height of the bullet and use it to calculate the bullet’s height after one-quarter of a second. (Round off to the nearest meter.)

8. An object is tossed into the air at an initial speed of $$30$$ feet per second from a rooftop $$10$$ feet high. Write a function that models the height of the object and use it to calculate the height of the object after $$1$$ second.

1. $$f ( - 3 ) = 27 ; f ( 0 ) = 6 ; f ( 10 ) = 196$$

3. $$h ( - 3 ) = 12 ; h ( 0 ) = 3 ; h ( 2 ) = - 13$$

5. (a) $$60$$ feet; (b) $$2$$ seconds; (c) $$4$$ seconds

7. $$h ( t ) = - 4.9 t ^ { 2 } + 340 t$$; at $$0.25$$ second, the bullet's height is about $$85$$ meters.

Exercise $$\PageIndex{2}$$

Perform the operations.

1. Given $$f ( x ) = 5 x ^ { 2 } - 3 x + 1$$ and $$g ( x ) = 2 x ^ { 2 } - x - 1$$, find $$( f + g ) ( x )$$.
2. Given $$f ( x ) = x ^ { 2 } + 3 x - 8$$ and $$g ( x ) = x ^ { 2 } - 5 x - 7$$, find $$( f - g ) ( x )$$.
3. Given $$f ( x ) = 3 x ^ { 2 } - x + 2$$ and $$g ( x ) = 2 x - 3$$, find $$( f \cdot g ) ( x )$$.
4. Given $$f ( x ) = 27 x ^ { 5 } - 15 x ^ { 3 } - 3 x ^ { 2 }$$ and $$g ( x ) = 3 x ^ { 2 }$$, find $$( f / g ) ( x )$$.
5. Given $$g ( x ) = x ^ { 2 } - x + 1$$, find $$g ( - 3 u )$$.
6. GIven $$g ( x ) = x ^ { 3 } - 1$$, find $$g ( x - 1 )$$.

1. $$( f + g ) ( x ) = 7 x ^ { 2 } - 4 x$$

3. $$( f \cdot g ) ( x ) = 6 x ^ { 3 } - 11 x ^ { 2 } + 7 x - 6$$

5. $$g ( - 3 u ) = 9 u ^ { 2 } + 3 u + 1$$

Exercise $$\PageIndex{3}$$

Given $$f ( x ) = 16 x ^ { 3 } - 12 x ^ { 2 } + 4 x , g ( x ) = x ^ { 2 } - x + 1$$, and $$h ( x ) = 4 x$$, find the following:

1. $$( g \cdot h ) ( x )$$
2. $$( f - g ) ( x )$$
3. $$( g + f ) ( x )$$
4. $$( f / h ) ( x )$$
5. $$( f \cdot h ) ( - 1 )$$
6. $$( g + h ) ( - 3 )$$
7. $$( g - f ) ( 2 )$$
8. $$( f / h ) \left( \frac { 3 } { 2 } \right)$$

1. $$( g \cdot h ) ( x ) = 4 x ^ { 3 } - 4 x ^ { 2 } + 4 x$$

3. $$( g + f ) ( x ) = 16 x ^ { 3 } - 11 x ^ { 2 } + 3 x + 1$$

5. $$( f \cdot h ) ( - 1 ) = 128$$

7. $$( g - f ) ( 2 ) = - 85$$

Exercise $$\PageIndex{4}$$

Factor our the greatest common factor (GCF).

1. $$2 x ^ { 4 } - 12 x ^ { 3 } - 2 x ^ { 2 }$$
2. $$18 a ^ { 3 } b - 3 a ^ { 2 } b ^ { 2 } + 3 a b ^ { 3 }$$
3. $$x ^ { 4 } y ^ { 3 } - 3 x ^ { 3 } y + x ^ { 2 } y$$
4. $$x ^ { 3 n } - x ^ { 2 n } - x ^ { n }$$

1. $$2 x ^ { 2 } \left( x ^ { 2 } - 6 x - 1 \right)$$

3. $$x ^ { 2 } y \left( x ^ { 2 } y ^ { 2 } - 3 x + 1 \right)$$

Exercise $$\PageIndex{5}$$

Factor by grouping.

1. $$2 x ^ { 3 } - x ^ { 2 } + 2 x - 1$$
2. $$3x ^ { 3 } - x ^ { 2 } - 6 x + 2$$
3. $$x ^ { 3 } - 5 x ^ { 2 } y + x y ^ { 2 } - 5 y ^ { 3 }$$
4. $$a ^ { 2 } b - a + a b ^ { 3 } - b ^ { 2 }$$
5. $$2 x ^ { 4 } - 4 x y ^ { 3 } + 2 x ^ { 2 } y ^ { 2 } - 4 x ^ { 3 } y$$
6. $$x ^ { 4 } y ^ { 2 } - x y ^ { 5 } + x ^ { 3 } y ^ { 4 } - x ^ { 2 } y ^ { 3 }$$

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

3. $$\left( x ^ { 2 } + y ^ { 2 } \right) ( x - 5 y )$$

5. $$2 x ( x - 2 y ) \left( x ^ { 2 } + y ^ { 2 } \right)$$

Exercise $$\PageIndex{6}$$

Factor the special binomials.

1. $$64 x ^ { 2 } - 1$$
2. $$9 - 100 y ^ { 2 }$$
3. $$x ^ { 2 } - 36 y ^ { 2 }$$
4. $$4 - ( 2 x - 1 ) ^ { 2 }$$
5. $$a ^ { 3 } b ^ { 3 } + 125$$
6. $$64 x ^ { 3 } - y ^ { 3 }$$
7. $$81 x ^ { 4 } - y ^ { 4 }$$
8. $$x ^ { 8 } - 1$$
9. $$x ^ { 6 } - 64 y ^ { 6 }$$
10. $$1 - a ^ { 6 } b ^ { 6 }$$

1. $$( 8 x + 1 ) ( 8 x - 1 )$$

3. $$( x + 6 y ) ( x - 6 y )$$

5. $$( a b + 5 ) \left( a ^ { 2 } b ^ { 2 } - 5 a b + 25 \right)$$

7. $$\left( 9 x ^ { 2 } + y ^ { 2 } \right) ( 3 x + y ) ( 3 x - y )$$

9. $$( x + 2 y ) \left( x ^ { 2 } - 2 x y + 4 y ^ { 2 } \right) ( x - 2 y ) \left( x ^ { 2 } + 2 x y + 4 y ^ { 2 } \right)$$

Exercise $$\PageIndex{7}$$

Factor.

1. $$x ^ { 2 } - 8 x - 48$$
2. $$x ^ { 2 } - 15 x + 54$$
3. $$x ^ { 2 } - 4 x - 6$$
4. $$x ^ { 2 } - 12 x y + 36 y ^ { 2 }$$
5. $$x ^ { 2 } + 20 x y + 75 y ^ { 2 }$$
6. $$- x ^ { 2 } + 5 x + 150$$
7. $$- 2 y ^ { 2 } + 20 y + 48$$
8. $$28 x ^ { 2 } + 20 x + 3$$
9. $$150 x ^ { 2 } - 100 x + 6$$
10. $$24 a ^ { 2 } - 38 a b + 3 b ^ { 2 }$$
11. $$27 u ^ { 2 } - 3 u v - 4 v ^ { 2 }$$
12. $$16 x ^ { 2 } y ^ { 2 } - 78 x y + 27$$
13. $$16 m ^ { 2 } + 72 m n + 81 n ^ { 2 }$$
14. $$4 x ^ { 2 } - 5 x + 20$$
15. $$25 x ^ { 4 } - 35 x ^ { 2 } + 6$$
16. $$2 x ^ { 4 } + 7 x ^ { 2 } + 3$$
17. $$x ^ { 6 } + 3 x ^ { 3 } y ^ { 3 } - 10 y ^ { 6 }$$
18. $$a ^ { 6 } - 8 a ^ { 3 } b ^ { 3 } + 15 b ^ { 6 }$$
19. $$x ^ { 2 n } - 2 x ^ { n } + 1$$
20. $$6 x ^ { 2 n } - x ^ { n } - 2$$

1. $$( x - 12 ) ( x + 4 )$$

3. Prime

5. $$( x + 5 y ) ( x + 15 y )$$

7. $$- 2 ( y - 12 ) ( y + 2 )$$

9. $$2 ( 15 x - 1 ) ( 5 x - 3 )$$

11. $$( 3 u + v ) ( 9 u - 4 v )$$

13. $$( 4 m + 9 n ) ^ { 2 }$$

15. $$\left( 5 x ^ { 2 } - 6 \right) \left( 5 x ^ { 2 } - 1 \right)$$

17. $$\left( x ^ { 3 } + 5 y ^ { 3 } \right) \left( x ^ { 3 } - 2 y ^ { 3 } \right)$$

19. $$\left( x ^ { n } - 1 \right) ^ { 2 }$$

Exercise $$\PageIndex{8}$$

Factor completely.

1. $$45 x ^ { 3 } - 20 x$$
2. $$12 x ^ { 4 } - 70 x ^ { 3 } + 50 x ^ { 2 }$$
3. $$- 20 x ^ { 2 } + 32 x - 3$$
4. $$- x ^ { 3 } y + 9 x y ^ { 3 }$$
5. $$24 a ^ { 4 } b ^ { 2 } + 3 a b ^ { 5 }$$
6. $$64 a ^ { 6 } b ^ { 6 } - 1$$
7. $$64 x ^ { 2 } + 1$$
8. $$x ^ { 3 } + x ^ { 2 } y - x y ^ { 2 } - y ^ { 3 }$$

1. $$5 x ( 3 x + 2 ) ( 3 x - 2 )$$

3. $$- ( 10 x - 1 ) ( 2 x - 3 )$$

5. $$3 a b ^ { 2 } ( 2 a + b ) \left( 4 a ^ { 2 } - 2 a b + b ^ { 2 } \right)$$

7. Prime

Exercise $$\PageIndex{9}$$

Solve by factoring.

1. $$9 x ^ { 2 } + 8 x = 0$$
2. $$x ^ { 2 } - 1 = 0$$
3. $$x ^ { 2 } - 12 x + 20 = 0$$
4. $$x ^ { 2 } - 2 x - 48 = 0$$
5. $$( 2 x + 1 ) ( x - 2 ) = 3$$
6. $$2 - ( x - 4 ) ^ { 2 } = - 7$$
7. $$( x - 6 ) ( x + 3 ) = - 18$$
8. $$( x + 5 ) ( 2 x - 1 ) = 3 ( 2 x - 1 )$$
9. $$\frac { 1 } { 2 } x ^ { 2 } + \frac { 2 } { 3 } x - \frac { 1 } { 8 } = 0$$
10. $$\frac { 1 } { 4 } x ^ { 2 } - \frac { 19 } { 12 } x + \frac { 1 } { 2 } = 0$$
11. $$x ^ { 3 } - 2 x ^ { 2 } - 24 x = 0$$
12. $$x ^ { 4 } - 5 x ^ { 2 } + 4 = 0$$

1. $$-\frac{8}{9} , 0$$

3. $$2,10$$

5. $$-1, \frac{5}{2}$$

7. $$0,3$$

9. $$- \frac { 3 } { 2 } , \frac { 1 } { 6 }$$

11. $$- 4,0,6$$

Exercise $$\PageIndex{10}$$

Find the roots of the given functions.

1. $$f ( x ) = 12 x ^ { 2 } - 8 x$$
2. $$g ( x ) = 2 x ^ { 3 } - 18 x$$
3. $$h ( t ) = - 16 t ^ { 2 } + 64$$
4. $$p ( x ) = 5 x ^ { 2 } - 21 x + 4$$

6.

7. The height in feet of an object dropped from the top of a $$16$$-foot ladder is given by $$h ( t ) = - 16 t ^ { 2 } + 16$$, where $$t$$ represents the time in seconds after the object has been dropped. How long will it take to hit the ground?

8. The length of a rectangle is $$2$$ centimeters less than twice its width. If the area of the rectangle is $$112$$ square centimeters, find its dimensions.

9. A triangle whose base is equal in measure to its height has an area of $$72$$ square inches. Find the length of the base.

10. A box can be made by cutting out the corners and folding up the edges of a sheet of cardboard. A template for a rectangular cardboard box of height $$2$$ inches is given.

What are the dimensions of a cardboard sheet that will make a rectangular box with volume $$240$$ cubic inches?

1. $$0 , \frac { 2 } { 3 }$$

3. $$\pm 2$$

5. $$- 9,0,6$$

7. $$1$$ second

9. $$12$$ inches

Exercise $$\PageIndex{11}$$

Solve or factor.

1. $$x ^ { 2 } - 25$$
2. $$x ^ { 2 } - 121 = 0$$
3. $$16 x ^ { 2 } - 22 x - 3 = 0$$
4. $$3 x ^ { 2 } - 14 x - 5$$
5. $$x ^ { 3 } - x ^ { 2 } - 2 x - 2$$
6. $$3 x ^ { 2 } = - 15 x$$

1. Factor; $$( x + 5 ) ( x - 5 )$$

3. Solve; $$- \frac { 1 } { 8 } , \frac { 3 } { 2 }$$

5. Factor; $$( x - 1 ) \left( x ^ { 2 } - 2 \right)$$

Exercise $$\PageIndex{12}$$

Find a polynomial equation with integer coefficients, given the solutions.

1. $$5, -2$$
2. $$\frac { 2 } { 3 } , - \frac { 1 } { 2 }$$
3. $$\pm \frac { 4 } { 5 }$$
4. $$\pm 10$$
5. $$-4,0,3$$
6. $$-8$$ double root

1. $$x ^ { 2 } - 3 x - 10 = 0$$

3. $$25 x ^ { 2 } - 16 = 0$$

5. $$x ^ { 3 } + x ^ { 2 } - 12 x = 0$$

Exercise $$\PageIndex{13}$$

State the restrictions and simplify.

1. $$\frac { 108 x ^ { 3 } } { 12 x ^ { 2 } }$$
2. $$\frac { 56 x ^ { 2 } ( x - 2 ) ^ { 2 } } { 8 x ( x - 2 ) ^ { 3 } }$$
3. $$\frac { 64 - x ^ { 2 } } { 2 x ^ { 2 } - 15 x - 8 }$$
4. $$\frac { 3 x ^ { 2 } + 28 x + 9 } { 81 - x ^ { 2 } }$$
5. $$\frac { x ^ { 2 } - 25 } { 5 x ^ { 2 } } \cdot \frac { 10 x ^ { 2 } - 15 x } { 2 x ^ { 2 } + 7 x - 15 }$$
6. $$\frac { 7 x ^ { 2 } - 41 x - 6 } { ( x - 7 ) ^ { 2 } } \cdot \frac { 49 - x ^ { 2 } } { x ^ { 2 } + x - 42 }$$
7. $$\frac { 28 x ^ { 2 } ( 2 x - 3 ) } { 4 x ^ { 2 } - 9 } \div \frac { 7 x } { 4 x ^ { 2 } - 12 x + 9 }$$
8. $$\frac { x ^ { 2 } - 10 x + 24 } { x ^ { 2 } - 8 x + 16 } \div \frac { 2 x ^ { 2 } - 13 x + 6 } { x ^ { 2 } + 2 x - 24 }$$

1. $$9 x ; x \neq 0$$

3. $$- \frac { x + 8 } { 2 x + 1 } ; x \neq - \frac { 1 } { 2 } , 8$$

5. $$\frac { x - 5 } { x } ; x \neq - 5,0 , \frac { 3 } { 2 }$$

7. $$\frac { 4 x ( 2 x - 3 ) ^ { 2 } } { 2 x + 3 } ; x \neq \pm \frac { 3 } { 2 } , 0$$

Exercise $$\PageIndex{14}$$

Perform the operations and simplify. Assume all variable expressions in the denominator are nonzero.

1. $$\frac { a ^ { 2 } - b ^ { 2 } } { 4 a ^ { 2 } b ^ { 2 } + 4 a b ^ { 3 } } \cdot \frac { 2 a b } { a ^ { 2 } - 2 a b + b ^ { 2 } }$$
2. $$\frac { a ^ { 2 } - 5 a b + 6 b ^ { 2 } } { a ^ { 2 } - 4 a b + 4 b ^ { 2 } } \div \frac { 9 b ^ { 2 } - a ^ { 2 } } { 3 a ^ { 3 } b - 6 a ^ { 2 } b ^ { 2 } }$$
3. $$\frac { x ^ { 2 } + x y + y ^ { 2 } } { 4 x ^ { 2 } + 3 x y - y ^ { 2 } } \cdot \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 3 } - y ^ { 3 } } \div \frac { x + y } { 12 x ^ { 2 } y - 3 x y ^ { 2 } }$$
4. $$\frac { x ^ { 4 } - y ^ { 4 } } { x ^ { 2 } - 2 x y + y ^ { 2 } } \div \frac { x ^ { 2 } - 4 x y - 5 y ^ { 2 } } { 10 x ^ { 3 } } \cdot \frac { 2 x ^ { 2 } - 11 x y + 5 y ^ { 2 } } { 2 x ^ { 3 } y + 2 x y ^ { 3 } }$$

1. $$\frac { 1 } { 2 b ( a - b ) }$$

3. $$\frac { 3 x y } { x + y }$$

Exercise $$\PageIndex{15}$$

Perform the operations and state the restrictions.

1. Given $$f ( x ) = \frac { 4 x ^ { 2 } + 39 x - 10 } { x ^ { 2 } + 3 x - 10 }$$ and $$g ( x ) = \frac { 2 x ^ { 2 } + 7 x - 15 } { x ^ { 2 } + 13 x + 30 }$$, find $$( f \cdot g ) ( x )$$.
2. Given $$f ( x ) = \frac { 25 - x ^ { 2 } } { 3 + x }$$ and $$g ( x ) = \frac { 9 - x ^ { 2 } } { 5 - x }$$, find $$( f \cdot g ) ( x )$$.
3. Given $$f ( x ) = \frac { 42 x ^ { 2 } } { 2 x ^ { 2 } + 3 x - 2 }$$ and $$g ( x ) = \frac { 14 x } { 4 x ^ { 2 } - 4 x + 1 }$$, find $$( f / g ) ( x )$$.
4. Given $$f ( x ) = \frac { x ^ { 2 } - 20 x + 100 } { x ^ { 2 } - 1 }$$ and $$g ( x ) = \frac { x ^ { 2 } - 100 } { x ^ { 2 } + 2 x + 1 }$$, find $$( f / g ) ( x )$$.
5. The daily cost in dollars of running a small business is given by $$C (x) = 150 + 45x$$ where $$x$$ represents the number of hours the business is in operation. Determine the average cost per hour if the business is in operation for $$8$$ hours in a day.
6. An electric bicycle manufacturer has determined that the cost of producing its product in dollars is given by the function $$C (n) = 2n^{2} + 100n + 2,500$$ where $$n$$ represents the number of electric bicycles produced in a day. Determine the average cost per bicycle if $$10$$ and $$20$$ are produced in a day.
7. Given $$f ( x ) = 3 x - 5$$, simplify $$\frac { f ( x + h ) - f ( x ) } { h }$$.
8. Given $$g ( x ) = 2 x ^ { 2 } - x + 1$$, simplify $$\frac { g ( x + h ) - g ( x ) } { h }$$.

1. $$( f \cdot g ) ( x ) = \frac { ( 4 x - 1 ) ( 2 x - 3 ) } { ( x - 2 ) ( x + 3 ) } ; x \neq - 10 , - 5 , - 3,2$$

3. $$( f / g ) ( x ) = \frac { 3 x ( 2 x - 1 ) } { x + 2 } ; x \neq - 2,0 , \frac { 1 } { 2 }$$

5. $$\ 63.75$$ per hour

7. $$3$$

Exercise $$\PageIndex{16}$$

State the restrictions and simplify.

1. $$\frac { 5 x - 6 } { x ^ { 2 } - 36 } - \frac { 4 x } { x ^ { 2 } - 36 }$$
2. $$\frac { 2 } { x } + 5 x$$
3. $$\frac { 5 } { x - 5 } + \frac { 1 } { 2 x }$$
4. $$\frac { x } { x - 2 } + \frac { 3 } { x + 3 }$$
5. $$\frac { 7 ( x - 1 ) } { 4 x ^ { 2 } - 17 x + 15 } - \frac { 2 } { x - 3 }$$
6. $$\frac { 5 } { x } - \frac { 19 x + 25 } { 2 x ^ { 2 } + 5 x }$$
7. $$\frac { x } { x - 5 } - \frac { 2 } { x - 3 } - \frac { 5 ( x - 3 ) } { x ^ { 2 } - 8 x + 15 }$$
8. $$\frac { 3 x } { 2 x - 1 } - \frac { x - 4 } { x + 4 } + \frac { 12 ( 2 - x ) } { 2 x ^ { 2 } + 7 x - 4 }$$
9. $$\frac { 1 } { t - 1 } + \frac { 1 } { ( t - 1 ) ^ { 2 } } - \frac { 1 } { t ^ { 2 } - 1 }$$
10. $$\frac { 1 } { t - 1 } - \frac { 2 t - 5 } { t ^ { 2 } - 2 t + 1 } - \frac { 5 t ^ { 2 } - 3 t - 2 } { ( t - 1 ) ^ { 3 } }$$
11. $$2 x ^ { - 1 } + x ^ { - 2 }$$
12. $$( x - 4 ) ^ { - 1 } - 2 x ^ { - 2 }$$

1. $$\frac { 1 } { x + 6 } ; x \neq \pm 6$$

3. $$\frac { 11 x - 5 } { 2 x ( x - 5 ) } ; x \neq 0,5$$

5. $$- \frac { 1 } { 4 x - 5 } ; x \neq \frac { 5 } { 4 } , 3$$

7. $$\frac { x - 5 } { x - 3 } ; x \neq 3,5$$

9. $$\frac { t ^ { 2 } + 1 } { ( t + 1 ) ( t - 1 ) ^ { 2 } } ; t \neq \pm 1$$

11. $$\frac { 2 x + 1 } { x ^ { 2 } } ; x \neq 0$$

Exercise $$\PageIndex{17}$$

Simplify. Assume that all variable expressions used as denominators are nonzero.

1. $$\frac { \frac { 1 } { 7 } + \frac { 1 } { x } } { \frac { 1 } { 49 } - \frac { 1 } { x ^ { 2 } } }$$
2. $$\frac { \frac { 1 } { 100 } - \frac { 1 } { x ^ { 2 } } } { \frac { 1 } { x } - \frac { 1 } { 10 } }$$
3. $$\frac { \frac { 3 } { x } - \frac { 1 } { x - 5 } } { \frac { 5 } { x + 2 } - \frac { 2 } { x } }$$
4. $$\frac { 1 - \frac { 12 } { x } + \frac { 35 } { x ^ { 2 } } } { 1 - \frac { 25 } { x ^ { 2 } } }$$
5. $$\frac { x - 4 x ^ { - 1 } } { 2 - 5 x ^ { - 1 } + 2 x ^ { - 2 } }$$
6. $$\frac { 8 x ^ { - 1 } + y ^ { - 1 } } { y ^ { - 2 } - 64 x ^ { - 2 } }$$

1. $$\frac { 7 x } { x - 7 }$$

3. $$\frac { ( x + 2 ) ( 2 x - 15 ) } { ( x - 5 ) ( 3 x - 4 ) }$$

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

Exercise $$\PageIndex{18}$$

Perform the operations and state the restrictions.

1. Given $$f ( x ) = \frac { 3 } { x - 3 }$$ and $$g ( x ) = \frac { x - 2 } { x + 2 }$$, find $$( f + g ) ( x )$$.
2. Given $$f ( x ) = \frac { 1 } { x ^ { 2 } + x }$$ and $$g ( x ) = \frac { 2 x } { x ^ { 2 } - 1 }$$, find $$( f + g ) ( x )$$.
3. Given $$f ( x ) = \frac { x - 3 } { x - 5 }$$ and $$g ( x ) = \frac { x ^ { 2 } - x } { x ^ { 2 } - 25 }$$, find $$( f - g ) ( x )$$.
4. Given $$f ( x ) = \frac { 11 x + 4 } { x ^ { 2 } - 2 x - 8 }$$ and $$g ( x ) = \frac { 2 x } { x - 4 }$$, find $$( f - g ) ( x )$$.

1. $$( f + g ) ( x ) = \frac { x ^ { 2 } - 2 x + 12 } { ( x - 3 ) ( x + 2 ) } ; x \neq - 2,3$$

3. $$( f - g ) ( x ) = \frac { 3 } { x + 5 } ; x \neq \pm 5$$

Exercise $$\PageIndex{19}$$

Solve.

1. $$\frac { 3 } { x } = \frac { 1 } { 2 x + 15 }$$
2. $$\frac { x } { x - 4 } = \frac { x + 8 } { x - 8 }$$
3. $$\frac { x + 5 } { 2 ( x + 2 ) } + \frac { x - 2 } { x + 4 } = 1$$
4. $$\frac { 2 x } { x - 5 } + \frac { 1 } { x + 1 } = 0$$
5. $$\frac { x + 1 } { x - 4 } + \frac { 4 } { x + 6 } = - \frac { 10 } { x ^ { 2 } + 2 x - 24 }$$
6. $$\frac { 2 } { x } - \frac { 12 } { 2 x + 3 } = \frac { 2 - 3 x ^ { 2 } } { 2 x ^ { 2 } + 3 x }$$
7. $$\frac { x + 7 } { x - 2 } - \frac { 9 } { x + 7 } = \frac { 81 } { x ^ { 2 } + 5 x - 14 }$$
8. $$\frac { x } { x + 5 } + \frac { 1 } { x - 4 } = \frac { 4 x - 7 } { x ^ { 2 } + x - 20 }$$
9. $$\frac { 2 } { 3 x - 1 } + \frac { x } { 2 x + 1 } = \frac { 2 ( 3 - 4 x ) } { 6 x ^ { 2 } + x - 1 }$$
10. $$\frac { x } { x - 1 } + \frac { 1 } { x + 1 } = \frac { 2 x } { x ^ { 2 } - 1 }$$
11. $$\frac { 2 x } { x + 5 } - \frac { 1 } { 2 x - 3 } = \frac { 4 - 7 x } { 2 x ^ { 2 } + 7 x - 15 }$$
12. $$\frac { x } { x + 4 } + \frac { 1 } { 2 x + 7 } = \frac { x + 8 } { 2 x ^ { 2 } + 15 x + 28 }$$
13. $$\frac { 1 } { t - 1 } - \frac { 2 } { 2 t + 1 } = \frac { 1 } { t - 2 } - \frac { 2 } { 2 t - 1 }$$
14. $$\frac { t - 1 } { t - 2 } - \frac { t - 2 } { t - 3 } = \frac { t - 3 } { t - 4 } - \frac { t - 4 } { t - 5 }$$
15. Solve for $$a$$ : $$\frac { 1 } { a } = \frac { 1 } { b } - \frac { 1 } { c }$$
16. Solve for $$y$$ : $$x = \frac { 3 y - 1 } { y - 5 }$$
17. A positive integer is $$4$$ less than another. If the reciprocal of the larger integer is subtracted from twice the reciprocal of the smaller, the result is $$\frac{1}{6}$$. Find the two integers.
18. If $$3$$ times the reciprocal of the larger of two consecutive odd integers is added to $$7$$ times the reciprocal of the smaller, the result is $$\frac{4}{3}$$. Find the integers.
19. If the reciprocal of the smaller of two consecutive integers is subtracted from three times the reciprocal of the larger, the result is $$\frac{3}{10}$$. Find the integers.
20. A positive integer is twice that of another. The sum of the reciprocals of the two positive integers is $$\frac{1}{4}$$. Find the two integers.

1. $$−9$$

3. $$−1, 4$$

5. $$−11, 0$$

7. $$Ø$$

9. $$−4$$

11. $$−\frac{3 }{2}$$

13. $$\frac{3}{4}$$

15. $$a = \frac { b c } { c - b }$$

17. $${8, 12}$$

19. $${5, 6}$$

Exercise $$\PageIndex{20}$$

Use algebra to solve the following applications.

1. Manuel traveled $$8$$ miles on the bus and another $$84$$ miles on a train. If the train was $$16$$ miles per hour faster than the bus, and the total trip took $$2$$ hours, what was the average speed of the train?
2. A boat can average $$10$$ miles per hour in still water. On a trip downriver, the boat was able to travel $$7.5$$ miles with the current. On the return trip, the boat was only able to travel $$4.5$$ miles in the same amount of time against the current. What was the speed of the current?
3. Susan can jog, on average, $$1 \frac{1}{2}$$ miles per hour faster than her husband Bill. Bill can jog $$10$$ miles in the same amount of time it takes Susan to jog $$13$$ miles. How fast, on average, can Susan jog?
4. In the morning, Raul drove $$8$$ miles to visit his grandmother and then returned later that evening. Because of traffic, his average speed on the return trip was $$\frac{1}{2}$$ that of his average speed that morning. If the total driving time was $$\frac{3}{4}$$ of an hour, what was his average speed on the return trip?
5. One pipe can completely fill a water tank in $$6$$ hours while another smaller pipe takes $$8$$ hours to fill the same tank. How long will it take to fill the tank to $$\frac{3}{4}$$ capacity if both pipes are turned on?
6. It takes Bill $$3$$ minutes longer than Jerry to fill an order. Working together they can fill $$15$$ orders in $$30$$ minutes. How long does it take Bill to fill an order by himself?
7. Manny takes twice as long as John to assemble a skateboard. If they work together, they can assemble a skateboard in $$6$$ minutes. How long would it take Manny to assemble the skateboard without John’s help?
8. Working alone, Joe can complete the yard work in $$30$$ minutes. It takes Mike $$45$$ minutes to complete work on the same yard. How long would it take them working together?

1. $$48$$ miles per hour

3. $$6.5$$ miles per hour

5. Approximately $$2.6$$ hours

7. $$18$$ minutes

Exercise $$\PageIndex{21}$$

Construct a mathematical model given the following:

1. $$y$$ varies directly as $$x$$, where $$y=30$$ when $$x=5$$.
2. $$y$$ varies inversely as $$x$$, where $$y=3$$ when $$x=-2$$.
3. $$y$$ is jointly proportional to $$x$$ and $$z$$, where $$y=-50$$ when $$x=-2$$ and $$z=5$$.
4. $$y$$ is directly proportional to the square of $$x$$ and inversely proportional to $$z$$, where $$y=-6$$ when $$x=2$$ and $$z=-8$$.
5. The distance an object in free fall varies directly with the square of the time that it has been falling. It is observed that an object falls $$36$$ feet in $$1 \frac{1}{2}$$ seconds. Find an equation that models the distance an object will fall, and use it to determine how far it will fall in $$2 \frac{1}{2}$$ seconds.
6. After the brakes are applied, the stopping distance $$d$$ of an automobile varies directly with the square of the speed $$s$$ of the car. If a car traveling $$55$$ miles per hour takes $$181.5$$ feet to stop, how many feet will it take to stop if it is moving $$65$$ miles per hour?
7. The weight of an object varies inversely as the square of its distance from the center of the Earth. If an object weighs $$180$$ lbs on the surface of the Earth (approximately $$4,000$$ miles from the center), then how much will it weigh at $$2,000$$ miles above the Earth’s surface?
8. The cost per person of renting a limousine varies inversely with the number of people renting it. If $$5$$ people go in on the rental, the limousine will cost $$112$$ per person. How much will the rental cost per person if $$8$$ people go in on the rental?
9. To balance a seesaw, the distance from the fulcrum that a person must sit is inversely proportional to his weight. If a $$52$$-pound boy is sitting $$3$$ feet away from the fulcrum, then how far from the fulcrum must a $$44$$-pound boy sit? Round to the nearest tenth of a foot.

1. $$y=6x$$

3. $$y=5xz$$

5. $$d=16t^{2}$$; $$100$$ feet

7. $$80$$ lbs

9. Approximately $$3.5$$ feet

## Sample Exam

Exercise $$\PageIndex{22}$$

Given $$f ( x ) = x ^ { 2 } - x + 4 , g ( x ) = 5 x - 1$$, and $$h ( x ) = 2 x ^ { 2 } + x - 3$$, find the following:

1. $$( g \cdot h ) ( x )$$
2. $$( h - f ) ( x )$$
3. $$( f + g ) ( - 1 )$$

1. $$( g \cdot h ) ( x ) = 10 x ^ { 3 } + 3 x ^ { 2 } - 16 x + 3$$

3. $$( f + g ) ( - 1 ) = 0$$

Exercise $$\PageIndex{23}$$

Factor.

1. $$x ^ { 3 } + 16 x - 2 x ^ { 2 } - 32$$
2. $$x ^ { 3 } - 8 y ^ { 3 }$$
3. $$x ^ { 4 } - 81$$
4. $$25 x ^ { 2 } y ^ { 2 } - 40 x y + 16$$
5. $$16 x ^ { 3 } y + 12 x ^ { 2 } y ^ { 2 } - 18 x y ^ { 3 }$$

2. $$( x - 2 y ) \left( x ^ { 2 } + 2 x y + 4 y ^ { 2 } \right)$$

4. $$( 5 x y - 4 ) ^ { 2 }$$

Exercise $$\PageIndex{24}$$

Solve

1. $$6 x ^ { 2 } + 24 x = 0$$
2. $$( 2 x + 1 ) ( 3 x + 2 ) = 12$$
3. $$( 2 x + 1 ) ^ { 2 } = 23 x + 6$$
4. Find a quadratic equation with integer coefficients given the solutions $$\left\{ \frac { 1 } { 2 } , - 3 \right\}$$.
5. Given $$f ( x ) = 5 x ^ { 2 } - x + 4$$, simplify $$\frac { f ( x + h ) - f ( x ) } { h }$$, where $$h \neq 0$$.

1. $$-4,0$$

3. $$-\frac{1}{4}, 5$$

5. $$10x+5h-1$$

Exercise $$\PageIndex{25}$$

Simplify and state the restrictions.

1. $$\frac { 4 x ^ { 2 } - 33 x + 8 } { x ^ { 2 } - 10 x + 16 } \div \frac { 16 x ^ { 2 } - 1 } { x ^ { 2 } - 4 x + 4 }$$
2. $$\frac { x - 1 } { x - 7 } + \frac { 1 } { 1 - x } - \frac { 2 ( x + 11 ) } { x ^ { 2 } - 8 x + 7 }$$

2. $$\frac { x + 2 } { x - 1 } ; x \neq 1,7$$

Exercise $$\PageIndex{26}$$

Assume all variable expressions in the denominator are nonzero and simplify.

1. $$\frac { \frac { 3 } { x } + \frac { 1 } { y } } { \frac { 1 } { y ^ { 2 } } - \frac { 9 } { x ^ { 2 } } }$$

Exercise $$\PageIndex{27}$$

Solve.

1. $$\frac { 6 x - 5 } { 3 x + 2 } = \frac { 2 x } { x + 1 }$$
2. $$\frac { 2 x } { x + 5 } - \frac { 1 } { 5 - x } = \frac { 2 x } { x ^ { 2 } - 25 }$$
3. Find the root of the function defined by $$f ( x ) = \frac { 1 } { x + 3 } - 4$$.
4. Solve for $$y$$: $$x = \frac { 4 y } { 3 y - 1 }$$

1. $$-\frac{5}{3}$$

3. $$-\frac{11}{4}$$

Exercise $$\PageIndex{28}$$

Use algebra to solve.

1. The height of an object dropped from a $$64$$-foot building is given by the function $$h (t) = −16t^{2} + 64$$, where $$t$$ represents time in seconds after it was dropped.
1. Determine the height of the object at $$\frac{3}{4}$$ of a second.
2. How long will it take the object to hit the ground?
2. One positive integer is $$3$$ units more than another. When the reciprocal of the larger is subtracted from twice the reciprocal of the smaller, the result is $$\frac{2}{9}$$. Find the two positive integers.
3. A light airplane can average $$126$$ miles per hour in still air. On a trip, the airplane traveled $$222$$ miles with a tailwind. On the return trip, against a headwind of the same speed, the plane was only able to travel $$156$$ miles in the same amount of time. What was the speed of the wind?
4. On the production line, it takes John $$2$$ minutes less time than Mark to assemble a watch. Working together they can assemble $$5$$ watches in $$12$$ minutes. How long does it take John to assemble a watch working alone?
5. Write an equation that relates $$x$$ and $$y$$, given that $$y$$ varies inversely with the square of $$x$$, where $$y = −\frac{1}{3}$$ when $$x = 3$$. Use it to find $$y$$ when $$x = \frac{1}{2}$$.
1. (1) $$55$$ feet; (2) $$2$$ seconds
3. $$22$$ miles per hour
5. $$y = - \frac { 3 } { x ^ { 2 } } ; y = - 12$$