### All Algebra 1 Resources

## Example Questions

### Example Question #401 : Polynomials

Which of the following is a factor of the polynomial ?

**Possible Answers:**

None of the other choices is correct.

**Correct answer:**

None of the other choices is correct.

To factor , we use the "reverse-FOIL".

, where and are two integers whose product is 24 and whose sum is 12. If we examine all of the factor pairs of 24, we note that the sum of each pair is as follows:

1 and 24: 25

2 and 12: 14

3 and 8: 11

4 and 6: 10

No factor pair of 24 has sum 12, so cannot be factored. The correct response is that none of the four binomials is correct.

### Example Question #4633 : Algebra 1

Which of the following is a prime polynomial?

**Possible Answers:**

The polynomials in all four of the other choices are prime.

**Correct answer:**

The sum of two squares is in general a prime polynomial unless a greatest common factor can be distributed out. is the sum of squares, and its terms do not have a GCF, so it is the prime polynomial.

Of the remaining choices:

is equal to ; as the difference of squares, it is factorable.

is equal to ; this fits the pattern of a perfect square quadratic trinomial, and is therefore factorable. is factorable for a similar reason.

### Example Question #4634 : Algebra 1

How many ways can a positive integer be written in the box to form a factorable polynomial?

**Possible Answers:**

Seven

Three

None

Two

One

**Correct answer:**

One

Let be the integer written in the box.

If is factorable, then its factorization is , where and . In other words, must be the positive difference of two numbers of a factor pair of 7. 7 has only one factor pair, 1 and 7, so the only possible value of is . The correct choice is one.

### Example Question #4635 : Algebra 1

Which of the following is a factor of ?

**Possible Answers:**

**Correct answer:**

can be factored by grouping, as follows:

cannot be factored further. Of the five choices, is the only factor.

### Example Question #4636 : Algebra 1

Which of the following is a prime polynomial?

**Possible Answers:**

**Correct answer:**

A quadratic trinomial of the form can be factored by splitting the middle term into two terms. The two coefficients must have sum and product .

In each case, a factor pair of must be examined. These pairs, along with their sums, are:

1 and 24: 25

2 and 12: 14

3 and 8: 11

4 and 6: 10

Each polynomial with one of these four integers as its linear coefficient can be factored. The odd one out is , since there is no factor pair of 24 whose sum is 20. This is the correct choice.

### Example Question #1 : How To Factor A Variable

Solve for , when :

**Possible Answers:**

**Correct answer:**

First, factor the numerator, which should be . Now the left side of your equation looks like

Second, cancel the "like" terms - - which leaves us with .

Third, solve for by setting the left-over factor equal to 0, which leaves you with

### Example Question #1 : How To Factor A Variable

Factor the following expression:

**Possible Answers:**

**Correct answer:**

Here you have an expression with three variables. To factor, you will need to pull out the greatest common factor that each term has in common.

Only the last two terms have so it will not be factored out. Each term has at least and so both of those can be factored out, outside of the parentheses. You'll fill in each term inside the parentheses with what the greatest common factor needs to be multiplied by to get the original term from the original polynomial:

### Example Question #4 : Simplifying Expressions

Factor the expression:

**Possible Answers:**

**Correct answer:**

To find the greatest common factor, we need to break each term into its prime factors:

Looking at which terms all three expressions have in common; thus, the GCF is . We then factor this out: .

### Example Question #1 : How To Factor A Variable

Factor the expression:

**Possible Answers:**

**Correct answer:**

To find the greatest common factor, we must break each term into its prime factors:

The terms have , , and in common; thus, the GCF is .

Pull this out of the expression to find the answer: .

### Example Question #2 : How To Factor A Variable

If , and and are distinct positive integers, what is the smallest possible value of ?

**Possible Answers:**

**Correct answer:**

Consider the possible values for (*x*, *y*):

(1, 100)

(2, 50)

(4, 25)

(5, 20)

Note that (10, 10) is not possible since the two variables must be distinct. The sums of the above pairs, respectively, are:

1 + 100 = 101

2 + 50 = 52

4 + 25 = 29

5 + 20 = 25, which is the smallest sum and therefore the correct answer.

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