# Review Sheet for Test 2

Math 101

7-27-2012

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) .

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 using long division.

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

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

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

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

6. (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 .

7. Simplify, cancelling any common factors:

(a) .

(b) .

(c) .

8. Simplify, cancelling any common factors:

(a) .

(b) .

• Note: " " means "divide".

(c) .

(d) .

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

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

# Solutions to the Review Sheet for Test 2

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) .

Using the rule ,

(d) .

Using the rule ,

(e) .

Using the rule ,

(f) .

Using FOIL, I get

(g) .

Using the form , I get

(h) .

3. Factor completely:

(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) .

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 using long division.

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

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

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

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) Compute the quotient and remainder when is divided by .

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

Divide by :

Thus,

6. (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.

For , . Thus, you can't cancel x's in to get 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 .

If , , and , then

Thus, is not in general equal to .

7. Simplify, cancelling any common factors:

(a) .

(b) .

(c) .

8. Simplify, cancelling any common factors:

(a) .

(b) .

(c) .

(d) .

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

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

Today is the tomorrow you worried about yesterday. Now you know why. - Anonymous