Simplifying rational expressions and restrictions

Simplifying Rational Expressions and Restrictions

What is a Rational Expression?

Before getting into simplifying rational expressions and their restrictions, it is important to understand what a rational expression is. Rational expressions are simply equations that feature a denominator that is also a function. They can be simple or widely complex. The presence of a function in the denominator makes rational expressions unique compared to other polynomials, as they also often feature what are called “restrictions” or restricted values. These values are restrictions on variables due to the fact that a denominator equal to zero is impossible – we cannot divide by zero because it is undefined. Below are a few examples of rational expressions, as well as some simple restricted values of rational expressions.

x3x2(x0)\frac{x}{3x^{2}} (x \neq 0)
3x7(x0) \frac{3}{x^{7}} (x \neq 0)
4x2+2x2(x2) \frac{4x^{2}+2}{x-2} (x \neq 2)

Oftentimes, when dealing with rational expressions, we first need to conduct what is known as simplification. That is, trying to make the polynomial in the numerator and the denominator as simple as possible before we try to solve.

How to Simplify Rational Expressions:

To simplify rational expressions, you first must be familiar with working with regular polynomial functions. So, before reading further, please check out our articles on common factors of polynomials, how to factor polynomials, how to factor complex polynomials, and the perfect square trinomial. Once you’ve mastered these topics, you’ll be more than ready to simplify expressions with rational functions!

Simplifying expressions with rational functions isn’t too different from working with polynomials. In fact, there are no new techniques you need to learn! It’s simply about applying techniques in a different way.

As always, the best way to master simplifying rational expressions is to do some practice problems.

Example 1:

For rational expression shown below, simplify as much as possible:

5xx28x+15\frac{5-x}{x^{2}-8x+15}

Step 1: Factor

The most common simplification you will have to make with rational expressions is factoring. It is for this reason that you should always be on the lookout for potential factors, both on the numerator and the denominator. In this case, we can factor the quadratic polynomial in the denominator

5xx28x+15=5x(x5)(x3)\frac{5-x}{x^{2}-8x+15} = \frac{5-x}{(x-5)(x-3)}

Step 2: Try to Cancel Out Terms

After factoring, the next most common strategy is to try to cancel out terms in either the numerator and/or the denominator. In this case, with simple algebra, we can cancel out a term from both the numerator and the denominator.

Since:

5 - x = -(x - 5)

Therefore:

(x5)(x5)(x3)\frac{-(x-5)}{(x-5)(x-3)}

Step 3: Write Final Answer

When you are sure no more simplifications can be made, be sure to write out your final, simplified, answer. After cancelling out the (x-5), the final answer is:

1x3\frac{-1}{x-3}

Example 2:

For rational expression shown below, simplify as much as possible:

9t316t3t2+4t\frac{9t^{3}-16t}{3t^{2}+4t}

Step 1: Factor

Factor the t out from the expression.

t(9t216)t(3t+4)\frac{t(9t^{2}-16)}{t(3t+4)}

Step 2: Try to Cancel Out Terms

We can cancel out t, leaving us with:

(9t216)(3t+4)\frac{(9t^{2}-16)}{(3t+4)}

Step 3: Look for More Simplifications

Notice we can make 9t2169t^{2} - 16 a difference of squares:

(3t)2(4)2(3t)^{2} - (4)^{2}

(3t+4)(3t4)(3t + 4)(3t - 4)

Now the expression becomes:

(3t+4)(3t4)3t+4\frac{(3t+4)(3t-4)}{3t+4}

Step 3: Write Final Answer

Cancelling out the (3t + 4), we get the final answer:

3t43t - 4

How to Find the Restricted Values:

Other than simplifying them, another common type of analysis you could be asked to do with regards to rational expressions is identify restricted values. Earlier, we looked at a few simple rational expressions with simple restrictions. However, things are rarely that simple. To find the restricted values of rational expressions, we often need to do some legwork first. Here are a couple examples.

Example 1:

For rational expression shown below, identify the restricted values:

5xx28x+15\frac{5-x}{x^{2}-8x+15}

To identify restricted values, all that needs to be done is figure out for what values of x the denominator equals zero. Because the denominator cannot equal to zero, these values for x are called our restricted values – as briefly discussed earlier. Thus, in this case, x28x+150x^{2} - 8x + 15 \neq 0.

In order to find these values for x, the first thing to do is factor. When we factor x28x+15x^{2} - 8x + 15, we get (x5)(x3)(x - 5)(x - 3).

Thus, since we want (x5)(x3)0(x - 5)(x - 3) \neq 0 to be true, (x5)(x - 5) and (x3)(x - 3) cannot equal to zero. Therefore, x5x \neq 5 or x3x \neq 3, and these are our restricted values.

Example 2:

For rational expression shown below, identify the restricted values:

9t316t3t2+4t\frac{9t^{3}-16t}{3t^{2}+4t}

Just like in the previous example, the denominator cannot equal to zero: 3t2+4t03t^{2} + 4t \neq 0. From here we look for the non-permissible, otherwise known as restricted, values.

When we factor the tt out of 3t2+4t3t^{2}+4t, we get t(3t+4)t(3t + 4) -- this cannot equal zero. Therefore:

t0t \neq 0

And:

3t+403t + 4 \neq 0

3t43t \neq -4

t43t \neq \frac{-4}{3}

So, the non-permissible values are 00 and 43\frac{-4}{3}.

And that’s all there is to it! For an excellent study tool, check out this rational expressions calculator here. Also, for further study, check out our articles on the rational function, the point of discontinuity, the vertical asymptote, and the horizontal asymptote.

Simplifying rational expressions and restrictions

A rational expression is a fraction that its numerator and/or denominator are polynomials. In this lesson, we will first learn how to find the non-permissible values of the variable in a rational expression. Then, we will how to simplify rational expressions.

Lessons

Notes:
\cdot multiplication rule: xaxb=xa+bx^a \cdot x^b=x^{a+b}

\cdot division rule: xaxb=xab\frac{x^a}{x^b}=x^{a-b}
  • 1.
    For each rational expression:
    i) determine the non-permissible values of the variable, then
    ii) simplify the rational expression
  • 2.
    For each rational expression:
    i) determine the non-permissible values of the variable, then
    ii) simplify the rational expression
  • 3.
    For each rational expression:
    i) determine the non-permissible values of the variable, then
    ii) simplify the rational expression
  • 4.
    The area of a rectangular window can be expressed as 4x2+13x+34{x^2} + 13x + 3, while its length can be expressed as 4x+14x + 1.
  • 5.
    For each rational expression:
    i) determine the non-permissible values for yy in terms of xx , then
    ii) simplify, where possible.
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Simplifying rational expressions and restrictions

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