Algebraic Concepts - ISEE Upper Level Mathematics Achievement
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Rewrite the polynomial in standard form:
Rewrite the polynomial in standard form:
The degree of a term of a polynomial with one variable is the exponent of that variable. The terms of a polynomial in standard form are written in descending order of degree. Therefore, we rearrange the terms by their exponent, from 5 down to 0, noting that we can rewrite the 
and constant terms with exponents 1 and 0, respectively:
The degree of a term of a polynomial with one variable is the exponent of that variable. The terms of a polynomial in standard form are written in descending order of degree. Therefore, we rearrange the terms by their exponent, from 5 down to 0, noting that we can rewrite the
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Simplify:
Simplify:
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Assume that 
. Simplify:
Assume that
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If 
, simplify:
If
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If 
, simplify:
If
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Define 
What is 
?
Define
What is
Substitute 
for 
in the definition:
Substitute
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Add:
Add:
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Simplify:
Simplify:
Start by reordering the expression to group like-terms together.
Combine like-terms to simplify.
Start by reordering the expression to group like-terms together.
Combine like-terms to simplify.
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Simplify:
Simplify:
We can expand the first term using FOIL:
Reorder the expression to group like-terms together.
Simplify by combining like-terms.
We can expand the first term using FOIL:
Reorder the expression to group like-terms together.
Simplify by combining like-terms.
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Simplify:
Simplify:
Expand each term by using FOIL:
Rearrange to group like-terms together.
Simplify by combining like-terms.
Expand each term by using FOIL:
Rearrange to group like-terms together.
Simplify by combining like-terms.
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Simplify:
Simplify:
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Which expression is equivalent to the expression 
?
Which expression is equivalent to the expression
The first step in simplifying this expression is to get the binomial out of the parentheses. It's important to note you cannot further simplify this binomial first, since there are no like terms in it.
Since you have a minus sign in front of the binomial, you need to flip the sign of both terms inside the parentheses to get rid of the parentheses (similar to distributing a negative one across the binomial):
Now you are able to combine like terms, making sure that exponents on the variables match exactly before you combine. The first and fourth terms are like terms, and the second and third terms are like terms.
To combine those terms, keep the variables and exponents the same and add up the coefficients. The first term has a coefficient of 
and the fourth term has a coefficient of 
, so they add up to a total of 
. The second term has a coefficient of 
and the third term has a coefficient of 
, so they add up to a total of 
.
This brings you to the final, simplified answer:
The first step in simplifying this expression is to get the binomial out of the parentheses. It's important to note you cannot further simplify this binomial first, since there are no like terms in it.
Since you have a minus sign in front of the binomial, you need to flip the sign of both terms inside the parentheses to get rid of the parentheses (similar to distributing a negative one across the binomial):
Now you are able to combine like terms, making sure that exponents on the variables match exactly before you combine. The first and fourth terms are like terms, and the second and third terms are like terms.
To combine those terms, keep the variables and exponents the same and add up the coefficients. The first term has a coefficient of
This brings you to the final, simplified answer:
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Simplify:
Simplify:
First, rewrite the problem so that like terms are next to each other.
Next, evaluate the terms in parentheses.
Rewrite the expression in simplest form.
First, rewrite the problem so that like terms are next to each other.
Next, evaluate the terms in parentheses.
Rewrite the expression in simplest form.
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Simplify:
Simplify:
First we rewrite the problem so that like terms are together.
Next we can place like terms in parentheses and evaluate the parentheses.
Now we rewrite the equation in simplest form.
First we rewrite the problem so that like terms are together.
Next we can place like terms in parentheses and evaluate the parentheses.
Now we rewrite the equation in simplest form.
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Simplify:
Simplify:
Combine like terms:
Combine like terms:
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Simplify:
Simplify:
Combine like terms:
Combine like terms:
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Simplify the following:
Simplify the following:
To simplify this expression, you need to combine like terms.
There are two terms with 
, one term with 
, and two terms without a variable.
This gives you the final answer:
Remember that when you subtract by a negative number, you are actually adding the inverse - thereby adding a positive number.
To simplify this expression, you need to combine like terms.
There are two terms with
This gives you the final answer:
Remember that when you subtract by a negative number, you are actually adding the inverse - thereby adding a positive number.
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Simplify the following:
Simplify the following:
In this problem, you need to combine like terms. Be very careful when combining like terms, since a few terms differ only by a few exponents. You have two terms with 
, two terms with 
, and one term with 
.
This leads to the final answer:
In this problem, you need to combine like terms. Be very careful when combining like terms, since a few terms differ only by a few exponents. You have two terms with
This leads to the final answer:
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Multiply:
Multiply:
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On Gina's swim team, 40 percent of the swimmers also play one other sport (cross country, soccer, or baseball) competitively. There are 20 people on her swim team. Of the swimmers who play one other sport, 2 of them participate in cross country, and twice as many people play soccer as those who play baseball. How many play soccer?
On Gina's swim team, 40 percent of the swimmers also play one other sport (cross country, soccer, or baseball) competitively. There are 20 people on her swim team. Of the swimmers who play one other sport, 2 of them participate in cross country, and twice as many people play soccer as those who play baseball. How many play soccer?
On Gina's swim team, 40 percent of the swimmers also play one other sport competitively. Given that there are 20 people on her swim team, 8 people play one other sport because 40 percent of 20 is 8.
If 2 swimmers run cross country that leaves 6 swimmers who play a different sport.
Given that twice as many play soccer as those who play baseball, it follows that 4 play soccer and 2 play baseball.
Thus, 4 is the correct answer.
On Gina's swim team, 40 percent of the swimmers also play one other sport competitively. Given that there are 20 people on her swim team, 8 people play one other sport because 40 percent of 20 is 8.
If 2 swimmers run cross country that leaves 6 swimmers who play a different sport.
Given that twice as many play soccer as those who play baseball, it follows that 4 play soccer and 2 play baseball.
Thus, 4 is the correct answer.
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