Polynomials - ACT Math
Card 0 of 756
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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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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What type of equation is the following?
(y + 2)(y + 4)(y + 1) = z
What type of equation is the following?
(y + 2)(y + 4)(y + 1) = z
The degree of a polynomial is the highest exponent of the terms.
Degree 0 – constant
Degree 1 – linear
Degree 2 – quadratic
Degree 3 – cubic
Degree 4 – quartic
Multiply out the equation:
(y + 2)(y + 4)(y + 1) = z
(y2 + 2y + 4y + 8)(y + 1) = z
y3 + 2y2 + 4y2 + 8y + y2 + 2y + 4y + 8 = z
The highest exponent is y3;therefore the equation is a degree 3 cubic.
The degree of a polynomial is the highest exponent of the terms.
Degree 0 – constant
Degree 1 – linear
Degree 2 – quadratic
Degree 3 – cubic
Degree 4 – quartic
Multiply out the equation:
(y + 2)(y + 4)(y + 1) = z
(y2 + 2y + 4y + 8)(y + 1) = z
y3 + 2y2 + 4y2 + 8y + y2 + 2y + 4y + 8 = z
The highest exponent is y3;therefore the equation is a degree 3 cubic.
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Find the degree of the polynomial:
Find the degree of the polynomial:
The degree of a polynomial is determined by the term with the highest degree. In this case that is 
, which has a degree of 
.
The degree of a polynomial is determined by the term with the highest degree. In this case that is
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What is the degree of the following polynomial?
What is the degree of the following polynomial?
The degree of a polynomial is determined by the term with the highest degree. In this case, the first term, 
, has the highest degree, 
. The degree of a term is calculated by adding the exponents of each variable in the term.
The degree of a polynomial is determined by the term with the highest degree. In this case, the first term,
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Expand:
Expand:
Expand:
Step 1: Use the distributive property
Step 2: Combine like terms
Expand:
Step 1: Use the distributive property
Step 2: Combine like terms
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Multiply these polynomials out and expand.
Multiply these polynomials out and expand.
The proper way to multiply polynomials is to essentially apply the distributive property continually.
So 
is really the same as 
, and so on.
Then, simply group together like terms to get the final answer:
The proper way to multiply polynomials is to essentially apply the distributive property continually.
So
Then, simply group together like terms to get the final answer:
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Two positive consecutive whole numbers that are even are both multiples of 
. The product of the two numbers is 
. What is the sum of the two integers?
Two positive consecutive whole numbers that are even are both multiples of
The question provides two positive whole numbers that are each multiples of 6 and also 6 numbers apart. This may be translated into variables, where the first number may be represented by "x" and the second number may be represented as "x+6" given that it is 6 numbers greater than x.
The problem provides the information that the product of these two numbers is 72. Using the new definitions for the numbers, this may be represented as:
(x)(x+6) = 72
This provides an equation that multiplies two polynomials (one with one term, which is a monomial and one with two terms, which is a binomial) and an ability to solve for what x (the first of the two numbers) may be.
Using FOIL, the result is 
. This may be rewritten as 
, which will provide the value of x after factoring.
, where (-6)(12) will provide the product of -72 and the sum of 12 and -6 will yield 6. The results indicate x as having two possible solutions: x=6 and x=-12. Returning back to the question, the goal is to find two positive numbers. This means that x=-12 is not a viable solution and that x=6 is. Now, revisiting the terms used to redefine the two numbers \[x and x+6\], x has been calculated. After substituting in the x value for the second term, the second number is 12 (6+6=12).
The final step of the problem is to solve for the sum of these two numbers:
6+12=18
The question provides two positive whole numbers that are each multiples of 6 and also 6 numbers apart. This may be translated into variables, where the first number may be represented by "x" and the second number may be represented as "x+6" given that it is 6 numbers greater than x.
The problem provides the information that the product of these two numbers is 72. Using the new definitions for the numbers, this may be represented as:
(x)(x+6) = 72
This provides an equation that multiplies two polynomials (one with one term, which is a monomial and one with two terms, which is a binomial) and an ability to solve for what x (the first of the two numbers) may be.
Using FOIL, the result is
The final step of the problem is to solve for the sum of these two numbers:
6+12=18
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Simplify:
Simplify:
When subtracting polynomials, it's helpful to remember that the "minus sign" gets distributed. It's as if the two polynomials are being added and a -1 is in front of the second polynomial.
This -1 will get multiplied to all the terms in the second polynomial that is being subtracted from the first, so it becomes 
. You may note that by multiplying by -1, every term in the polynomial switches its original sign. The problem then becomes:
From here, in order to simplify, because there's no equal sign, it can be deduced that we are not working toward a solution for x. The original problem is presented as an expression so an expression as an answer will be expected. In order to work towards a final simplified expression, like terms must be collected. This will provide the final answer.
When subtracting polynomials, it's helpful to remember that the "minus sign" gets distributed. It's as if the two polynomials are being added and a -1 is in front of the second polynomial.
This -1 will get multiplied to all the terms in the second polynomial that is being subtracted from the first, so it becomes
From here, in order to simplify, because there's no equal sign, it can be deduced that we are not working toward a solution for x. The original problem is presented as an expression so an expression as an answer will be expected. In order to work towards a final simplified expression, like terms must be collected. This will provide the final answer.
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Simplify:
Simplify:
Begin by distributing the subtraction of the second term in this question:
Now, you merely need to combine like terms:
Begin by distributing the subtraction of the second term in this question:
Now, you merely need to combine like terms:
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The expression 
is equivalent to which of the following?
The expression
To answer this question, we must distribute the 
to the rest of the variables 
, 
, and 
that are within the brackets.
To distribute a variable or number, you multiply that value with every other value within the brackets or parentheses. So, for this data:
We then simplify the expression by combining the variables we are multiplying together into expressions. For this data:
Be sure to keep all of your operations the same within the problem itself, unless the number being distributed is negative, which will then switch the signs with the brackets from positive to negative or negative to positive.
Therefore, our answer is 
.
To answer this question, we must distribute the
To distribute a variable or number, you multiply that value with every other value within the brackets or parentheses. So, for this data:
We then simplify the expression by combining the variables we are multiplying together into expressions. For this data:
Be sure to keep all of your operations the same within the problem itself, unless the number being distributed is negative, which will then switch the signs with the brackets from positive to negative or negative to positive.
Therefore, our answer is
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Solve the equation 
Solve the equation
To answer this question, we are solving for the values of 
that make this equation true.
To this, we need to get 
on a side by itself so we can evaluate it. To do this, we first add 
to both sides so that we can then begin to deal with the 
value. So, for this data:
can also be written as 
. Therefore:
Now we can divide both sides by 
and find the value of 
.
Therefore, the answer to this question is 
To answer this question, we are solving for the values of
To this, we need to get
Now we can divide both sides by
Therefore, the answer to this question is
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If 
, then what does 
equal?
If
To solve this equation, we substitute 
in for every instance of 
seen in the original equation 
.
Therefore the new equation would read
Now we must square the expression 
. To do this, you must multiply the expression by itself. Therefore:
We must now plug in our new value for 
into our original equation in place of 
.
Now we must distribute the 
into 
. To do this, you multiply each expression within the parenthesis by 
:
Therefore, our answer is 
.
To solve this equation, we substitute
Therefore the new equation would read
Now we must square the expression
We must now plug in our new value for
Now we must distribute the
Therefore, our answer is
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What is the value of the coefficient in front of the term that includes 
in the expansion of 
?
What is the value of the coefficient in front of the term that includes
Using the binomial theorem, the term containing the _x_2 _y_7 will be equal to
(2_x_)2(–y)7
=36(–4_x_2 _y_7)= -144_x_2_y_7
Using the binomial theorem, the term containing the _x_2 _y_7 will be equal to
(2_x_)2(–y)7
=36(–4_x_2 _y_7)= -144_x_2_y_7
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Give the coefficient of 
in the binomial expansion of 
.
Give the coefficient of
If the expression 
is expanded, then by the binomial theorem, the 
term is
or, equivalently, the coefficient of 
is
Therefore, the 
coefficient can be determined by setting
If the expression
or, equivalently, the coefficient of
Therefore, the
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Give the coefficient of 
in the binomial expansion of 
.
Give the coefficient of
If the expression 
is expanded, then by the binomial theorem, the 
term is
or, equivalently, the coefficient of 
is
Therefore, the 
coefficient can be determined by setting
:
If the expression
or, equivalently, the coefficient of
Therefore, the
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