# SAT II Math II : Operations with Polynomials

## Example Questions

### Example Question #3 : Simplifying Polynomials

Subtract the expressions below.

None of the other answers are correct.

Explanation:

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 #12 : Simplifying Expressions

Divide by .

Explanation:

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  ?

None of the other responses gives a correct answer.

None of the other responses gives a correct answer.

Explanation:

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.

Divide:

Explanation:

Divide termwise:

### Example Question #3 : Operations With Polynomials

Factor:

The polynomial is prime.

Explanation:

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

where .

### Example Question #1 : Operations With Polynomials

Factor completely:

The polynomial is prime.

Explanation:

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: