# College Algebra : Review and Other Topics

## Example Questions

### Example Question #1 : Factoring Polynomials

Factor the polynomial

Possible Answers:

Correct answer:

Explanation:

can be looked as . When A=1, as it does in this case, we can ignore it. So now we need to look at factors of "C" that add up to "B."

Factors of 20 are:

1   20

2   10

4   5

Of these three options, the 4 & 5 will add to 9, so we write

### Example Question #371 : College Algebra

Factor the polynomial

Possible Answers:

Correct answer:

Explanation:

needs to be seen as . Then we need to ask what factors of "C" will equal "B" if one of them is multiplied by "A"

So first thing is to find factors of "C," luckily 5 only has 2:

1 and 5

We need to write an equation that uses either 5, 1 and a "*2" to equal -9.

Now we can write out our factors knowing that we will be using -5,2,and 1.

(__X+___)(___X+____)

from the work above, we know we have to multiply the 2 and the -5, so they need to be in opposite factors

(2X+__)(X-5)

that only leaves one space for the 1

### Example Question #372 : College Algebra

Factor the polynomial

Possible Answers:

Correct answer:

Explanation:

needs to be seen as . Then you need to ask what factors of "C" will equal "B" when added together.

C=-18, factors of -18 are:

-1,18

-2,9

-3,6

1,-18

2,-9

3,-6

Of those factors, only -3,6 will give us "B", which in this case, is "3"

so

becomes

### Example Question #11 : Factoring Polynomials

Factor completely:

Possible Answers:

Correct answer:

Explanation:

Set , and, consequently, . Substitute to form a quadratic polynomial in :

Factor this trinomial by finding two numbers whose product is 8 and whose sum is . Through trial and error, these numbers can be found to be and , so

Substitute  back for :

The first binomial is the difference of squares; the second is prime since 8 is not a perfect square. Thus, the final factorization is

### Example Question #14 : Factoring Polynomials

Factor completely:

Possible Answers:

Correct answer:

Explanation:

Set , and, consequently, . Substitute to form a quadratic polynomial in :

Factor this trinomial by finding two numbers whose product is 8 and whose sum is . Through trial and error, these numbers can be found to be and , so

Substitute  back for :

Both factors are the difference of perfect cubes and can be factored further as such using the appropriate pattern:

### Example Question #12 : Factoring Polynomials

Factor:

Possible Answers:

Correct answer:

Explanation:

Step 1: Break down  into factors...

Step 2: Find two numbers that add or subtract to .

We will choose  and .

Step 3: Look at the equation and see which number needs to change sign..

According to the middle term,  must be negative.

So, the factors are  and .

Step 4: Factor by re-writing the solutions in the form:

So...

### Example Question #1 : Rational Expressions

Rationalize the following fraction:

Possible Answers:

Correct answer:

Explanation:

Rationalize the following fraction:

To rationalize a denominator, we will multiply the top and bottom of the fraction by the denominator.

And we have our answer

### Example Question #1 : Rational Expressions

Simplify the following:

Possible Answers:

Correct answer:

Explanation:

First we need to factor both polynomials.

Becomes

Now we cancel out any variables that are in BOTH the numerator and denominator. Remember that if a group of variables/numbers are inside parenthesis, they are considered a single term.

The common term in this case is , removing that from the equation gives us

### Example Question #371 : College Algebra

Simplify the following

Possible Answers:

Correct answer:

Explanation:

The first step is to factor both polynomials:

Becomes

Now we cancel out any terms that are in both the numerator and denominator. Remember that any variables/numbers that are in parenthesis are considered a single term.

In this case, the common term is

Once we remove that, we are left with:

### Example Question #1 : Rational Expressions

Simplify the following:

Possible Answers:

Correct answer:

Explanation:

The first step is to factor both polynomials

becomes

Now we cancel out any terms that are in both the numerator and denominator. Remember that any variables/numbers that are in parenthesis are considered a single term.

In this case, the common term is

Simplified, the equation is: