Operations with Polynomials - SAT Math

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 .

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.

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 termwise:

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

where .

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 :

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.

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 .

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.

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 termwise:

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

where .

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 :

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.

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 .

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.

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 termwise:

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

where .

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 :