6.4 Add/Subtract Rational Expressions with Binomial and Trinomial Denominators

Adding and subtracting RE with binomial and trinomial denominators involves using the skills of factoring (when needed), identifying the LCD, and the NPV. Adding and subtracting RE with binomial and trinomial denominators uses the same characteristics as a monomial denominator, the goal is to have the same/common denominators in order to add or subtract the numerators. Here is a simple expression to simplify,

As you can see the denominators in this expression are the same which means we can just add the numerators which is 4x + 6x and the answer is

We can also determine the NPV (non-permissible value), which is -5 (x≠-5). NPV is important because it makes us become aware of which values are not allowed into the denominator, because if the x in the denominator equals -5 then the denominator would equal to 0 (-5+5=0) and we know that a number cannot be divided by zero. If you put any number divided by 0 in your calculator it would should up as ERROR.

Here are the steps for adding and subtracting rational expressions:

1. Factor the expression
2. Identify the LCD (Lowest Common Denominator)
3. State the NPV (Non-Permissible Value)
4. Rewrite fraction over the LCD

Here is another simple example but with different denominators:

We cannot factor but we can identify the LCD which is:

Since the denominator on the left fraction is missing an (x) we must times (x) by the numerator and denominator and with the right fraction it is missing a (z) so you times the numerator and denominator by z so the denominators would be the same. It is important to remember that what happens to the denominator must happen with the numerator as well.

Then, state the NPV which are x ≠ 0 and z ≠ 0

Since the denominators are the same we can combine them and make it one fraction (over the LCD)

Here is another equation:

As you can see the denominators are different so they must be the same in order to simplify, you can do the cross method where the left denominator multiplies with the right numerator and the right denominator multiplies with the left numerator, after multiplying the numerators remember to multiply the denominators as well, it should look something like this:

Note how the denominators are the same, now we can add m to 5m and -4 to 15 which makes our answer to:

and the NPV is m ≠ -3, 4

Here is an equation that requires factoring to start:

First, we factor the denominators in both fractions

We can then switch the right fraction denominator into -x(x-6), this would make finding and calculating the LCD easier and simpler

Then identify the LCD which is x(x-6)(x+6) and the NPV are x ≠ 6,-6,0

Remember that subtracting a negative fraction is the same as adding a positive fraction so we can rewrite the equation as

Then we apply the LCD to the fractions

Then the numerator becomes 2x+6, which we can factor to 2(x+3)

We want to factor the numerator because in some cases, you can actually simplify a part of the expression in the numerator with the denominator which can help you fully simplify your expression.

Here is one more binomial expression that involves expanding trinomials:

Since we cannot factor we can identify the LCD which is (t-4)(t-5). So in order for the LCD to become (t-4)(t-5), is on the left fraction you must multiply both numerator and denominator by (t-5), it is (t-5) because the numerator already has the    (t-4) and the LCD is (t-4)(t-5) same with the other fraction but instead multiplying by (t-5) it is (t-4). The NPV is t ≠ 4,5

Then the numerator would be t²-2t-15 – (t²-6t+8), it is very important that you keep the trinomial in brackets because of the negative symbol, because if it is just t²-2t-15 – t²-6t+8, then that is wrong because the negative only applied to t². If there are brackets then the negative would apply to every number so t²-2t-15 – (t²-6t+8) would turn to t²-2t-15 – t²+6t-8,

next we can collect like terms which makes it 4t-23 so the answer would be:

Here is an example that include a trinomial in its denominator:

We can first factor x²-2x-8, (you will need to acquire the skill of factoring trinomials in order to correctly simplifying adding/subtracting rational expressions)

Here is the factor form of x²-2x-8, also I personally like to put the 3 over 1 so I can visualize multiplying the denominator.

Then we find the LCD which is (x-4)(x+2), which the right fraction already has so you just multiply (x-4)(x+2) to the left fraction.

The NPV would be x ≠ -2,4

Then you can expand the left fraction numerator

Then we can write the expression over the LCD and combine like terms

Then factor numerator

Here is one last example that seems intimidating but if you use the same skills as before it is quite simple

First factor all the trinomials in the denominator

Then we can change the middle fraction to addition by multiplying the denominator by negative 1

Then apply LCD to the numerators according to each numerator. The LCD is (x+3)(x-2)(x+2). So the NPV is x≠ -3.-2, 2

Expand numerators

Collect like-terms and write fraction over LCD

Factor numerator

As you can see there is (x+2) and (x+3) in the numerator and denominator so they can cancel out

Which makes our final answer to:

NPV = x ≠ -3.-2, 2

It is important to find your non-permissible values after you find your LCD because as you can see the values can disappear when fully factored

At the end if you want to verify if your fully simplified version matches your original expression just pick any number(s) that isn’t a NPV and plug it into the variables in the expression and if you get the same answer in both simplified and non simplified version then you simplified it correctly

In conclusion, adding and subtracting rational expression with binomial and trinomial denominators is quite easy as long as you have the foundation skills of doing them, such as, factoring trinomials/binomials, identifying the LCD, stating the non-permissible value, and a general sense of combing like-terms regarding the numerator and denominator.