Multiplying and Dividing Rational Expressions - Algebra II
Card 0 of 124
(9_x_2 – 1) / (3_x_ – 1) =
(9_x_2 – 1) / (3_x_ – 1) =
It's much easier to use factoring and canceling than it is to use long division for this problem. 9_x_2 – 1 is a difference of squares. The difference of squares formula is a_2 – b_2 = (a + b)(a – b). So 9_x_2 – 1 = (3_x + 1)(3_x – 1). Putting the numerator and denominator together, (9_x_2 – 1) / (3_x_ – 1) = (3_x_ + 1)(3_x_ – 1) / (3_x_ – 1) = 3_x_ + 1.
It's much easier to use factoring and canceling than it is to use long division for this problem. 9_x_2 – 1 is a difference of squares. The difference of squares formula is a_2 – b_2 = (a + b)(a – b). So 9_x_2 – 1 = (3_x + 1)(3_x – 1). Putting the numerator and denominator together, (9_x_2 – 1) / (3_x_ – 1) = (3_x_ + 1)(3_x_ – 1) / (3_x_ – 1) = 3_x_ + 1.
Compare your answer with the correct one above
Simplify:
Simplify:
The numerator is equivalent to
The denominator is equivalent to
Dividing the numerator by the denominator, one gets
The numerator is equivalent to
The denominator is equivalent to
Dividing the numerator by the denominator, one gets
Compare your answer with the correct one above
Simplify:
Simplify:
Factor both the numerator and the denominator which gives us the following:
After cancelling 
we get
Factor both the numerator and the denominator which gives us the following:
After cancelling
Compare your answer with the correct one above
For all values 
, which of the following is equivalent to the expression above?
For all values 
First, factor the numerator. We need factors that multiply to 
and add to 
.
We can plug the factored terms into the original expression.
Note that 
appears in both the numerator and the denominator. This allows us to cancel the terms.
This is our final answer.
First, factor the numerator. We need factors that multiply to
We can plug the factored terms into the original expression.
Note that
This is our final answer.
Compare your answer with the correct one above
What is the slant asymptote of 
?
What is the slant asymptote of
To find the slant asymptote, we have to divide the numerator by the denominator and see the equation of the line that we get.
Our long division problem ends up looking like this:
(We can ignore the remainder because it doesn't sufficiently impact the equation of the asymptote, especially as 
approaches infinity.)
Thus, the equation of the slant asymptote is 
.
To find the slant asymptote, we have to divide the numerator by the denominator and see the equation of the line that we get.
Our long division problem ends up looking like this:
(We can ignore the remainder because it doesn't sufficiently impact the equation of the asymptote, especially as
Thus, the equation of the slant asymptote is
Compare your answer with the correct one above
Which is a simplified form of 
?
Which is a simplified form of
Compare your answer with the correct one above
Expand:
Expand:
To evaluate the expression, we will need to conduct the FOIL method on the first two polynomials and then use the distributive property to reach a final answer. Therefore:
which equals
Using the distributive property, we obtain:
which equals
To evaluate the expression, we will need to conduct the FOIL method on the first two polynomials and then use the distributive property to reach a final answer. Therefore:
which equals
Using the distributive property, we obtain:
which equals
Compare your answer with the correct one above
Expand:
Expand:
This problem will involve using the FOIL method to combine the first two parenthetical terms and then the distributive property to combine what is left. However, we can save time if we recognize that the first two parentheses are in form 
, with 
and 
. We can therefore combine these two parentheses in form 
, and therefore:
Now we can use FOIL to find that:
which gives us a final answer of
This problem will involve using the FOIL method to combine the first two parenthetical terms and then the distributive property to combine what is left. However, we can save time if we recognize that the first two parentheses are in form
Now we can use FOIL to find that:
which gives us a final answer of
Compare your answer with the correct one above
Simplify the following expression: 
Simplify the following expression:
Our first step in this problem would be to distribute the exponent on the second term, which makes the expression become:
We then multiply like terms, remembering that multiplying like terms with exponents means we will add the exponent, so the expression becomes:
Our first step in this problem would be to distribute the exponent on the second term, which makes the expression become:
We then multiply like terms, remembering that multiplying like terms with exponents means we will add the exponent, so the expression becomes:
Compare your answer with the correct one above
Evaluate the following expression:
Evaluate the following expression:
To divide monomials, we subtract the exponents of the like terms. Therefore:
and
Therefore:
To divide monomials, we subtract the exponents of the like terms. Therefore:
and
Therefore:
Compare your answer with the correct one above
Find the remainder after dividing by this binomial.
Find the remainder after dividing by this binomial.
Either using synthetic division by 2 or using x=2 in the remainder theorem are 2 short-cuts to performing the long division of this polynomial.
Either using synthetic division by 2 or using x=2 in the remainder theorem are 2 short-cuts to performing the long division of this polynomial.
Compare your answer with the correct one above
Simplify:
Simplify:
In order to solve this equation, we must first simplify it so that we can cancel common factors between the numerator and the denominator.
In the above equation, we can first factor a 
from 
. This gives us:
This is easier for us to factor. In order to factor this, we need to see which factors of 
have a sum of 
. This turns out to be 
and 
. Therefore, we can simplify this expression into:
Next, we need to simplify 
.
This is a difference of perfect squares. Therefore, its factors are 
.
Now we need to simplify 
.
This is a perfect square trinomial. Therefore, this simplifies in the form 
. Note that this is negative since in order for the middle term to be negative, the sign of 
must be negative as well.
Finally, we have to simplify 
.
To factor this, we need to see what multiples of 
(the first term, 
, multiplied by the third term, 
) have a sum of positive 
. This turns out to be positive 
plus a negative 
. Since our first term is 
, we need to determine which of our factors is a multiple of 
. We can see that this is only 
, which means that our factors will be positive 
and negative 
. Therefore, when we simplify our expression, we get a result of
Now our expression looks like
The 
in the numerator cancels with the 
in the denominator, the 
in the numerator cancels with the 
in the denominator, and one of the 
factors in the numerator cancels with the 
in the denominator. This gives us our solution of:
In order to solve this equation, we must first simplify it so that we can cancel common factors between the numerator and the denominator.
In the above equation, we can first factor a
This is easier for us to factor. In order to factor this, we need to see which factors of
Next, we need to simplify
This is a difference of perfect squares. Therefore, its factors are
Now we need to simplify
This is a perfect square trinomial. Therefore, this simplifies in the form
Finally, we have to simplify
To factor this, we need to see what multiples of
Now our expression looks like
The
Compare your answer with the correct one above
Simplify.
Simplify.
a. Like when dividing fractions, change the division sign to multiplication and use the reciprocal of the divisor.
b. Factor the trinomials in the numerator of both terms.
c. Cancel any common factors between the numerators and denominators.
This will leave: 
d. Multiply to simplify.
a. Like when dividing fractions, change the division sign to multiplication and use the reciprocal of the divisor.
b. Factor the trinomials in the numerator of both terms.
c. Cancel any common factors between the numerators and denominators.
This will leave:
d. Multiply to simplify.
Compare your answer with the correct one above
Simplify: 
Simplify:
There is a common factor in the numerator. Pull out the common factor and rewrite the numerator.
Factorize the denominator.
Cancel the 
term in the numerator and denominator.
The answer is: 
There is a common factor in the numerator. Pull out the common factor and rewrite the numerator.
Factorize the denominator.
Cancel the
The answer is:
Compare your answer with the correct one above
Multiply:
Multiply:
First factor the numerators and denominators of the two fractions. This allows us to re-write the original problem like this:
Now we can cancel terms that appear on both the top and the bottom, since they will divide to be a factor of 
. This means we can can cancel the top and bottom
, the top and bottom 
, and the top and bottom 
. This leaves us with the following answer:
First factor the numerators and denominators of the two fractions. This allows us to re-write the original problem like this:
Now we can cancel terms that appear on both the top and the bottom, since they will divide to be a factor of
Compare your answer with the correct one above
Multiply:
Multiply:
First, completely factor everything that can possibly be factored. This includes both numerators and the second denominator:
Now we can cancel everything that appears both on the top and the bottom, since it will divide to be a factor of 
:
We can simplify this by multiplying 
and 
.
This leaves us with the following answer:
First, completely factor everything that can possibly be factored. This includes both numerators and the second denominator:
Now we can cancel everything that appears both on the top and the bottom, since it will divide to be a factor of
We can simplify this by multiplying
This leaves us with the following answer:
Compare your answer with the correct one above
First, completely factor all 4 quadratics:
Now we can cancel all factors that appear on both the top and the bottom, because those will divide to a factor of 
. We quickly realize that all of the factors can be crossed off. This means that all of the factors have been divided to 
. This leves us with the following answer:
First, completely factor all 4 quadratics:
Now we can cancel all factors that appear on both the top and the bottom, because those will divide to a factor of
Compare your answer with the correct one above
First, combine the top two fractions. The common denominator between the two is 
Therefore, you just have to offset the first fraction so that it becomes 
. Then, combine the numerators to get 
. So at this point, we have: 
. This is essentially a dividing fractions problem. When we divide fractions, we have to make the second fraction its reciprocal (flip it!) and then multiply the two. 
. The 
's cross out so your final answer is: 
.
First, combine the top two fractions. The common denominator between the two is
Compare your answer with the correct one above
I would first start by simplifying the numerator by getting rid of the negative exponents: 
. Then, combine the denominator fractions into one fraction: 
. At this point, we're dividing fractions so we have to multiply by the reciprocal of the second fraction: 
. Multiply straight across to get: 
. Make sure it can't be simplified (it can't)!
I would first start by simplifying the numerator by getting rid of the negative exponents:
Compare your answer with the correct one above
Find the quotient of these rational expressions: 
Find the quotient of these rational expressions:
When you divide by a fraction you must multiply by its reciprocal to get the correct quotient.
Factor where able:
Cancel like terms:
When you divide by a fraction you must multiply by its reciprocal to get the correct quotient.
Factor where able:
Cancel like terms:
Compare your answer with the correct one above