Multiplying and Dividing Rational Expressions - Algebra II

The numerator is equivalent to

The denominator is equivalent to

Dividing the numerator by the denominator, one gets

Factor both the numerator and the denominator which gives us the following:

After cancelling we get

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.

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 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

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

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:

To divide monomials, we subtract the exponents of the like terms. Therefore:

and

Therefore:

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.

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:

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.

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:

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, 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 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, 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: .

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)!

When you divide by a fraction you must multiply by its reciprocal to get the correct quotient.

Factor where able:

Cancel like terms: