Math 101

These problems are provided to help you study. The presence of a
problem on this handout does not imply that there *will* be a
similar problem on the test. And the absence of a topic does not
imply that it *won't* appear on the test.

1. Simplify the expression, using only positive exponents in your answer. Assume that all variables represent positive quantities.

(a) .

(b) .

(c) .

(d) .

(e) .

2. Multiply the polynomials:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

(i) .

3. Factor completely:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

(i) .

(j) .

(k) .

(l) .

(m) .

(n) .

(o) .

(p) .

(q) .

(r) .

(s) .

(t) .

(u) .

(v) .

(w) .

(x) .

(y) .

(z) .

(aa) .

(bb) .

(cc) .

(dd) .

(ee) .

(ff) .

(gg) .

(hh) .

(ii) .

(jj) .

4. Solve the equation by factoring:

(a) .

(b) .

(c) .

(d) .

(e) .

5. (a) Divide by .

(b) Divide by .

6. (a) Divide by using long division.

(b) Find the quotient and remainder when is divided by .

(c) Compute the quotient and remainder when is divided by .

(d) Factor completely, given that is one of the factors.

7. (a) Show that it is *not* valid to cancel x's in to get by giving a specific value of
x for which is not equal to 5.

(b) Show that is not always the same as by giving specific values of a, b, and c for which is not equal to .

8. Simplify, cancelling any common factors:

(a) .

(b) .

(c) .

9. Simplify, cancelling any common factors:

(a) .

(b) .

(c) .

10. Combine the fractions into a single fraction and simplify:

(a) .

(b) .

(c) .

(d) .

(e) .

11. Simplify, cancelling any common factors:

(a) .

(b) .

(c) .

(d) .

- Hint: Factor everything you can factor, then multiply the top and bottom of the big fraction to clear the denominators of the little fractions.

12. (a) Solve .

(b) Solve .

(c) Solve .

(d) Solve .

(e) Solve .

(f) Solve .

13. If a number is added to the top and the bottom of , you get . What is the number?

14. Pipe A can fill a tank in 3 hours. If pipes A and B work together, they can fill 5 tanks in 6 hours. How long would it take for pipe B to fill a tank by itself?

15. Calvin can eat 800 Brussels sprouts in 4 hours. Phoebe can eat 600 Brussels sprouts in 5 hours. It takes Bonzo 3 hours to eat 240 Brussels sprouts. How long will it take them to eat 600 Brussels sprouts if they work together?

16. Calvin can eat 396 tacos in 6 hours. Calvin and Phoebe, eating together, can eat 324 tacos in 3 hours. Phoebe and Bonzo, eating together, can eat 456 tacos in 4 hours. How long will it take Bonzo to eat 576 tacos by himself?

17. The numerator of a fraction is 8 less than the denominator. If 7 is added to the top and the bottom of the fraction, the new fraction is equal to . Find the original fraction.

18. A river flows at a constant rate of 1.6 miles per hour. Calvin rows 32 miles downstream (with the current) in 5 hours less than it takes him to row the same distance upstream (against the current). What is Calvin's rowing speed in still water?

19. Calvin drives at an average speed that is 16 miles per hour faster than Bonzo's average speed. Bonzo takes 3 hours longer than Calvin to drive 288 miles. What is Bonzo's average speed?

1. Simplify the expression, using only positive exponents in your answer. Assume that all variables represent positive quantities.

(a) .

(b) .

(c) .

(d) .

(e) .

(a)

(b)

(c)

(d)

(e)

2. Multiply the polynomials:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

(i) .

(a)

(b)

(c)

In the problems that follow, you can multiply directly using "FOIL", you can use a multiplication formula (if you know it), or you can use grid multiplication.

(d)

Note: You could also use the rule .

(e)

(f)

(g)

(h)

(i)

3. Factor completely:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

(i) .

(j) .

(k) .

(l) .

(m) .

(n) .

(o) .

(p) .

(q) .

(r) .

(s) .

(t) .

(u) .

(v) .

(w) .

(x) .

(y) .

(z) .

(aa) .

(bb) .

(cc) .

(dd) .

(ee) .

(ff) .

(gg) .

(hh) .

(ii) .

(jj) .

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

(i) .

I'll use factoring by grouping:

(j) .

I'll use factoring by grouping:

(k) .

(l) .

(m) .

(n) .

(o) .

(p) .

(q) .

(r) .

(s) .

(t) .

(u) .

(v) .

(w) .

(x) .

(y) .

(z) .

(aa) .

Reminder: The formulas you need for the next few problems are:

(bb) .

(cc) .

(dd) .

(ee) .

(ff) .

In the next few problems, I'll use * factoring by
grouping*.

(gg) .

(hh) .

(ii) .

(jj) .

4. Solve the equation by factoring:

(a) .

(b) .

(c) .

(d) .

(e) .

(a)

Therefore, or .

(b)

gives , and the other factors give and . Therefore , , or .

(c)

gives .

If , then , which has no real solutions.

The only solution is .

(d) You need to get 0 on one side of the equation to use factoring to solve.

Therefore, or .

(e)

gives , so . The other two factors give and .

Therefore, , , or .

5. (a) Divide by .

(b) Divide by .

(a)

(b)

6. (a) Find the quotient and remainder when is divided by .

(b) Find the quotient and remainder when is divided by .

(c) Find the quotient and remainder when is divided by .

(d) Factor completely, given that is one of the factors.

(a)

(b)

(Since is missing an " " term, I put in " " as a place holder.)

(c)

Since there's only one term on the bottom, it's easier to break the fraction up into pieces than to do the long division.

(d) Divide by :

Thus,

7. (a) Show that it is *not* valid to cancel x's in to get by giving a specific value
of x for which is not equal to 5.

(b) Show that is not always the same as by giving specific values of a, b, and c for which is not equal to .

(a) For , . Thus, you can't cancel x's in to get 5.

(b) If , , and , then

Thus, is not in general equal to .

8. Simplify, cancelling any common factors:

(a) .

(b) .

(c) .

(a)

(b)

(c)

9. Simplify, cancelling any common factors:

(a) .

(b) .

(c) .

(a)

(b)

(c)

10. Combine the fractions into a single fraction and simplify:

(a) .

(b) .

(c) .

(d) .

(e) .

(a)

(b)

(c)

(d)

(e)

11. Simplify, cancelling any common factors:

(a) .

(b) .

(c) .

(d) .

(a)

(b)

(c)

(d)

12. (a) Solve .

(b) Solve .

(c) Solve .

(d) Solve .

(e) Solve .

(f) Solve .

(a)

Check:

The solution is .

(b)

Plugging into results in division by zero. Therefore, there are no solutions.

(c)

Check: When ,

The solution is .

(d)

Check: When ,

However, causes division by 0 in the original equation.

Therefore, the only solution is .

(e)

The possible solutions are and .

Check: For ,

For ,

The solutions are and .

(f)

The possible solutions are and .

Check: If ,

If ,

The solutions are and .

13. If a number is added to the top and the bottom of , you get . What is the number?

Let n be the number to be added.

If a number is added to the top and the bottom of , you get :

Clear the fractions and solve:

Check: When ,

The number is 11.

14. Pipe A can fill a tank in 3 hours. If pipes A and B work together, they can fill 5 tanks in 6 hours. How long would it take for pipe B to fill a tank by itself?

Let x be pipe A's rate in tanks per hour, and let y be pipe B's rate in tanks per hour.

The first equation says , so .

The third equation says . Substitute and solve for y:

The second equation says . Substitute into ; this gives , so hours.

15. Calvin can eat 800 Brussels sprouts in 4 hours. Phoebe can eat 600 Brussels sprouts in 5 hours. It takes Bonzo 3 hours to eat 240 Brussels sprouts. How long will it take them to eat 600 Brussels sprouts if they work together?

Let x be Calvin's rate in sprouts per hour, let y be Phoebe's rate in sprouts per hour, and let z be Bonzo's rate in sprouts per hour.

The first equation says , so .

The second equation says , so .

The third equation says , so .

The last equation says . Substitute , , and into :

It takes 1.5 hours.

16. Calvin can eat 396 tacos in 6 hours. Calvin and Phoebe, eating together, can eat 324 tacos in 3 hours. Phoebe and Bonzo, eating together, can eat 456 tacos in 4 hours. How long will it take Bonzo to eat 576 tacos by himself?

Let x be Calvin's rate in tacos per hour, let y be Phoebe's rate in tacos per hour, and let z be Bonzo's rate in tacos per hour.

The first row says , so .

The second row says . Plug in and solve for y:

The third row says . Plug in and solve for z:

The fourth row says . Plug in and solve for t:

It takes Bonzo 8 hours.

17. The numerator of a fraction is 8 less than the denominator. If 7 is added to the top and the bottom of the fraction, the new fraction is equal to . Find the original fraction.

Let be the original fraction.

The numerator is 8 less than the denominator, so

If 7 is added to the top and the bottom, the result equals :

Plug into :

Multiply by to clear denominators:

Solve for d:

Hence, . The fraction is .

18. A river flows at a constant rate of 1.6 miles per hour. Calvin rows 32 miles downstream (with the current) in 5 hours less than it takes him to row the same distance upstream (against the current). What is Calvin's rowing speed in still water?

Let x be Calvin's rowing speed in still water. His downstream rate is , while his downstream rate is .

Let t be the time it takes Calvin to row upstream. Then it takes him (that is, 5 hours less) to row downstream.

The equations are

Solve the second equation for t:

Plug this into and solve for x:

Since Calvin's speed can't be negative, I must have miles per hour.

19. Calvin drives at an average speed that is 16 miles per hour faster than Bonzo's average speed. Bonzo takes 3 hours longer than Calvin to drive 288 miles. What is Bonzo's average speed?

Let x be Bonzo's speed, and let t be the time it takes Calvin to drive 288 miles.

From the table, I get the equations

gives . Plugging this into and solving for x, I get

The solutions to the last equation are and . Speed can't be negative. Hence, Bonzo's speed is 32 miles per hour.

*If you don't like something, change it. If you can't change it,
change your attitude.* - *Maya Angelou*

Copyright 2020 by Bruce Ikenaga