### All SAT II Math II Resources

## Example Questions

### Example Question #1 : How To Multiply Trinomials

Multiply the expressions:

**Possible Answers:**

**Correct answer:**

You can look at this as the sum of two expressions multiplied by the difference of the same two expressions. Use the pattern

,

where and .

To find , you use the formula for perfect squares:

,

where and .

Substituting above, the final answer is .

### Example Question #1 : Expanding Expressions And Foil

Which of the following values of would make

a prime polynomial?

**Possible Answers:**

**Correct answer:**

A polynomial of the form whose terms do not have a common factor, such as this, can be factored by rewriting it as such that and ; the grouping method can be used on this new polynomial.

Therefore, for to be factorable, must be the sum of the two integers of a factor pair of . We are looking for a value of that is *not* a sum of two such factors.

The factor pairs of 96, along with their sums, are:

1 and 96 - sum 97

2 and 48 - sum 50

3 and 32 - sum 35

4 and 24 - sum 28

6 and 16 - sum 22

8 and 12 - sum 20

Of the given choices, only 30 does not appear among these sums; it is the correct choice.

### Example Question #2 : Expanding Expressions And Foil

How many of the following are *prime* factors of the polynomial ?

(A)

(B)

(C)

(D)

**Possible Answers:**

Three

One

Two

Four

None

**Correct answer:**

Two

making this polynomial the difference of two cubes.

As such, can be factored using the pattern

so

(A) and (C) are both factors, but not (B) or (D), so the correct response is two.

### Example Question #8 : Intermediate Single Variable Algebra

Subtract the expressions below.

**Possible Answers:**

None of the other answers are correct.

**Correct answer:**

Since we are only adding and subtracting (there is no multiplication or division), we can remove the parentheses.

Regroup the expression so that like variables are together. Remember to carry positive and negative signs.

For all fractional terms, find the least common multiple in order to add and subtract the fractions.

Combine like terms and simplify.

### Example Question #1 : Operations With Polynomials

Divide by .

**Possible Answers:**

**Correct answer:**

First, set up the division as the following:

Look at the leading term in the divisor and in the dividend. Divide by gives ; therefore, put on the top:

Then take that and multiply it by the divisor, , to get . Place that under the division sign:

Subtract the dividend by that same and place the result at the bottom. The new result is , which is the new dividend.

Now, is the new leading term of the dividend. Dividing by gives 5. Therefore, put 5 on top:

Multiply that 5 by the divisor and place the result, , at the bottom:

Perform the usual subtraction:

Therefore the answer is with a remainder of , or .

### Example Question #1 : Operations With Polynomials

Which of the following is a prime factor of ?

**Possible Answers:**

None of the other responses gives a correct answer.

**Correct answer:**

None of the other responses gives a correct answer.

can be seen to fit the pattern

:

where

can be factored as , so

, making this the difference of squares, so it can be factored as follows:

Therefore,

The polynomial has only two prime factors, each squared, neither of which appear among the choices.

### Example Question #2 : Operations With Polynomials

Divide:

**Possible Answers:**

**Correct answer:**

Divide termwise:

### Example Question #1 : Operations With Polynomials

Factor:

**Possible Answers:**

The polynomial is prime.

**Correct answer:**

can be rewritten as and is therefore the difference of two cubes. As such, it can be factored using the pattern

where .

### Example Question #4 : Operations With Polynomials

Factor completely:

**Possible Answers:**

The polynomial is prime.

**Correct answer:**

Since both terms are perfect cubes , the factoring pattern we are looking to take advantage of is the sum of cubes pattern. This pattern is

We substitute for and 8 for :

### Example Question #5 : Operations With Polynomials

Factor completely:

**Possible Answers:**

The polynomial is prime.

**Correct answer:**

The polynomial is prime.

Since the first term is a perfect cube, the factoring pattern we are looking to take advantage of is the difference of cubes pattern. However, 225 is *not* a perfect cube of an integer , so the factoring pattern cannot be applied. No other pattern fits, so the polynomial is a prime.