Polynomials - ACT Math
Card 0 of 20
Simplify the following expression.
Simplify the following expression.
Line up each expression vertically. Then combine like terms to solve.
____________________
Thus, the final solution is 
.
Line up each expression vertically. Then combine like terms to solve.
____________________
Thus, the final solution is
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What is the value of 
when 
What is the value of
In adding 
to both sides:
. . .and adding 
to both sides:
. . .the variables are isolated to become:
After dividing both sides by 
, the equation becomes:
In adding
. . .and adding
. . .the variables are isolated to become:
After dividing both sides by
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Add the following polynomials:
Add the following polynomials:
This is a problem where elimination can be help you save a little time. You can eliminate options quickly by simplifying one power at a time and comparing your work with the answer choices.
To begin, reorder the problem so that all like terms are next to each other. When doing so, keep an eye on your signs so that you don't accidentally make a mistake.
From here, combine each pair of terms. As you do so, compare your work with the answer choices.
Eliminate any answer choices that have a different 
term.
Eliminate any answer choices that have a different 
term.
Eliminate any answer choices that have a different x term.
Eliminate any answer choices that have a different constant term.
Once you put all of your solutions together, the correct answer looks like this:
This is a problem where elimination can be help you save a little time. You can eliminate options quickly by simplifying one power at a time and comparing your work with the answer choices.
To begin, reorder the problem so that all like terms are next to each other. When doing so, keep an eye on your signs so that you don't accidentally make a mistake.
From here, combine each pair of terms. As you do so, compare your work with the answer choices.
Once you put all of your solutions together, the correct answer looks like this:
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Choose the answer which best simplifies the following expression:
Choose the answer which best simplifies the following expression:
To solve this problem simply remove the parentheses and add the like terms:
To solve this problem simply remove the parentheses and add the like terms:
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Add 
and 
.
Add
To add the trinomials, simply eliminate the parentheses and add like terms.
To add the trinomials, simply eliminate the parentheses and add like terms.
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Like terms can be added together: 
is added to 
, 
is added to 
, and 
is added to 
. The resulting answer choice that is correct is 
.
Like terms can be added together:
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Choose the answer which best simplifies the following expression:
Choose the answer which best simplifies the following expression:
To simplify, simply remove the parentheses and combine like terms:
To simplify, simply remove the parentheses and combine like terms:
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Choose the answer which best simplifies the following expression:
Choose the answer which best simplifies the following expression:
To simplify, remove parentheses and combine like terms:
To simplify, remove parentheses and combine like terms:
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Choose the answer which best simplifies the following expression:
Choose the answer which best simplifies the following expression:
To simplify, remove parentheses and combine like terms:
Note that adding or subtracting a zero to the end of this equation is unnecessary.
To simplify, remove parentheses and combine like terms:
Note that adding or subtracting a zero to the end of this equation is unnecessary.
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What is 
equal to?
What is
1. Factor the numerator:
2. Factor the denominator:
3. Divide the factored numerator by the factored denominator:
You can cancel out the 
from both the numerator and the denominator, leaving you with:
1. Factor the numerator:
2. Factor the denominator:
3. Divide the factored numerator by the factored denominator:
You can cancel out the
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Simplify:
Simplify:
In order to divide these polynomials, you need to first factor them.
and
Now, the expression becomes 
so,
In order to divide these polynomials, you need to first factor them.
and
Now, the expression becomes
so,
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Simplify the following division of polynomials:
Simplify the following division of polynomials:
The leading term of the numerator is one exponent higher than the leading term of the denominator. Thus, we know the result of the division is going to be somewhere close to 
. We can separate the fraction out like this:
The first term is easily seen to be 
, which is equal to 
. The second term can also be written out as:
and combining these, we get our final answer,
The leading term of the numerator is one exponent higher than the leading term of the denominator. Thus, we know the result of the division is going to be somewhere close to
The first term is easily seen to be
and combining these, we get our final answer,
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Divide:
Divide:
can be divided using long division.
The set up would look very similar to the division of real numbers, such as when we want to divide 10 by 2 and the answer is 5.
The first step after setting up the "division house" is to see what the first term in the outer trinomial needs to multiplied by to match the 
in the house. In this case, it's 
. 
will be multiplied across the other two terms in the outer trinomial and the product will be subtracted from the expression inside the division house. The following steps will take place in the same way.
While we could continue to divide, it would require the use of fraction exponents that would make the answer more complicated. Therefore, the term in red will be the remainder. Because this remainder is still subject to be being divided by the trinomial outside of the division house, we will make the remainder part of the final answer by writing it in fraction form:
Therefore the final answer is
The set up would look very similar to the division of real numbers, such as when we want to divide 10 by 2 and the answer is 5.
The first step after setting up the "division house" is to see what the first term in the outer trinomial needs to multiplied by to match the
While we could continue to divide, it would require the use of fraction exponents that would make the answer more complicated. Therefore, the term in red will be the remainder. Because this remainder is still subject to be being divided by the trinomial outside of the division house, we will make the remainder part of the final answer by writing it in fraction form:
Therefore the final answer is
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Simplify the expression:
Simplify the expression:
Once simplified, (x+1) appears on both the numerator and denominator, meaning we can cancel out both of them.
Which gives us:
Once simplified, (x+1) appears on both the numerator and denominator, meaning we can cancel out both of them.
Which gives us:
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Factor the polynomial: 2x2 + ab – 2b – ax2
Factor the polynomial: 2x2 + ab – 2b – ax2
2x2 + ab – 2b – ax2 = 2x2 – 2b – ax2 + ab
Rearrange terms
= (2x2 – 2b) + (- ax2 + ab) Group
= 2(x2 – b) + a(-x2 + b) Factor each group
= 2(x2 – b) – a(x2 – b) Factor out -a
= (x2 – b) (2 – a) Factor out x2 – b
2x2 + ab – 2b – ax2 = 2x2 – 2b – ax2 + ab
Rearrange terms
= (2x2 – 2b) + (- ax2 + ab) Group
= 2(x2 – b) + a(-x2 + b) Factor each group
= 2(x2 – b) – a(x2 – b) Factor out -a
= (x2 – b) (2 – a) Factor out x2 – b
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Which of the following is a factor of the polynomial 3y2+14y-24?
Which of the following is a factor of the polynomial 3y2+14y-24?
The polynomial factors to (3y-4)(y+6). (y+6) is the only one of those two options available as an answer.
The polynomial factors to (3y-4)(y+6). (y+6) is the only one of those two options available as an answer.
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Which of the following is a factor of the polynomial 
?
Which of the following is a factor of the polynomial
Factor the polynomial by choosing two values that when FOILed will sum to the middle coefficient, 3, and multiply to 2. These two numbers are 1 and 2.
Only (x +1) is one of the choices listed.
Factor the polynomial by choosing two values that when FOILed will sum to the middle coefficient, 3, and multiply to 2. These two numbers are 1 and 2.
Only (x +1) is one of the choices listed.
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Factor the polynomial : _x_3 + 27
Factor the polynomial : _x_3 + 27
First, write as a sum of the cubes,
_x_3 + 33, then factor : (x + 3) (x_2 – 3_x + 32). Apply the exponent:
(x + 3) (x_2 – 3_x + 9)
First, write as a sum of the cubes,
_x_3 + 33, then factor : (x + 3) (x_2 – 3_x + 32). Apply the exponent:
(x + 3) (x_2 – 3_x + 9)
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Which of the following expressions is a factor of this polynomial?
Which of the following expressions is a factor of this polynomial?
The polynomial factors into the following expression:
Therefore, the answer is 
The polynomial factors into the following expression:
Therefore, the answer is
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Which of the following is equivalent to 
?
Which of the following is equivalent to
You have two options for a problem like this. On the one hand, you can merely FOIL the answers until you find one that equals the value given in the equation. The other option is to factor it adequately from the beginning. This is not too hard. Since the middle term is so small, you know that the numeric portions of your factors will have to be nearly equal. Since we know that 
, this is very easy! Given that the last value is a negative number, you know that your two groups need to be a combination of addition and subtraction.
Thus, your answer is: 
You have two options for a problem like this. On the one hand, you can merely FOIL the answers until you find one that equals the value given in the equation. The other option is to factor it adequately from the beginning. This is not too hard. Since the middle term is so small, you know that the numeric portions of your factors will have to be nearly equal. Since we know that
Thus, your answer is:
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