As math students move into 8th grade, the math they are faced with is as intriguing as it is challenging. No longer are children locked into the idea that math is just about adding, subtracting, and multiplying whole numbers on the number line.
Rational numbers and rational expressions are parts of the math curriculum that are indeed connected, but that we use in very different ways.
Understanding rational numbers are straightforward enough. Formally, we can say that a rational number is a number that can be in the form p/q where p and q are integers and q does not equal zero.
In simple terms, this just means that rational numbers are numbers that are made by dividing one integer by another integer.
When we read the definition of rational expressions, however, we need to make sure we are up-to-date on our grade 7 polynomial lessons. Rational expressions are expressions that are rations of two polynomials.
Here is an example of a rational expression:
We can quickly remember that both rational expressions and rational numbers are fractions because “rational” contains the word “ratio”. This should point our brains toward fractions.
Rational Functions And Lowest Terms
We know that rational expressions are just like fractions, but with polynomials instead of integers.
A rational function is the ratio of two polynomials, where the bottom polynomial does not equal zero. Here is how we can write this out:
f(x) = P(x)/Q(x)
Q(x) cannot be zero (and anywhere that Q(x)=0 is undefined)
When we are faced with a function like this, we are usually asked to find its roots. To find the roots of a rational expression, we need to find the roots of the top polynomial, once we have put the rational expression into its lowest terms.
As we know from our earlier years of math class, if we want to simplify fractions, we want to get the fraction into its lowest terms. This is when the top and the bottom of the fraction have no common factors.
Similarly, when we are talking about rational expressions, these expressions are in their lowest terms when the top and bottom have no common factors.
If our rational expression is (x3+3x2)/2x it is not in its lowest terms, because we can divide the top and the bottom by x.
If we do this, we are left with (x2+3x)/2, which is in its lowest terms because the top and the bottom share no common factors.
Therefore, we now know that to find the roots of a rational expression, we must:
- Reduce the rational expression to its lowest terms
- Find the roots of the top polynomial
Proper Rational Expressions And Improper Rational Expressions
We can see that, as we fly through this introduction to rational expressions, there are many different aspects of this topic that we need to wrap our heads around. Another that we need to mention is the fact that rational expressions can be proper or improper.
Darling with regular fractions, we know that a proper fraction is when the numerator is smaller than the denominator, and an improper fraction is when the numerator is larger than or equal to the denominator.
What makes rational expressions proper or improper is the degree (or the largest exponent of x). Let’s look at some examples and identify the degree in some different polynomial expressions.
4x → the degree is one
4x3 − x + 3 → the degree is three
Now that we know this, we will better understand the definitions of proper and improper rational expressions.
A rational expression is proper if the degree of the top is less than the degree of the bottom.
A rational expression is improper if the degree of the top is greater than or equal to the degree of the bottom.
As we find out more about our rational expressions, we can dig deeper and establish even more interesting facts about these expressions. Later in this topic, students will be introduced to asymptotes and other interesting topics.
OMC 8th Grade Math Courses
Rational expressions are one topic that may seem daunting at first, but by relying on previously acquired knowledge and using skills that are applicable across different areas of math, kids can ensure that they are able to explain the difference between proper and improper rational expressions.
At OMC, we offer students the opportunity to enjoy tailored classes, entirely focused on reinforcing previous lessons, plugging any gaps that may exist in their knowledge, and giving them experience practicing math in examination environments. With our expert team of qualified professionals, your child will have the opportunity to excel in math, regardless of their age, level, or previous experience with the subject.