Introduction to Polynomials
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You have undoubtedly dealt with polynomials but just did not know that they were polynomials. If you’ve ever had to distribute and combine like terms, that was most likely done with polynomials.
In short, a polynomial is an Algebraic expression with Whole Number exponents. That means that there might be fractions, but the denominator will never have a variable. Polynomial itself is a generic term, translating roughly to many numbers. We will distinguish between three types of polynomial, and everything else we’ll just call a polynomial.
Let’s talk about some common language. This way you won’t be lost when reading directions!
Terms are parts of a polynomial separated by addition, subtraction, or an equal sign (for equations).
Coefficients are the numbers multiplying with a variable. For example, in the monomial example above, the coefficient is 3.
Leading Coefficient is the coefficient for the term with the highest exponent.
Degree is the value of the polynomial’s largest exponent (for one variable polynomials).
Descending Order is a polynomial organized by alphabetical order, then exponent size.
Like Terms are terms that have the exact same combination of variables and exponents.
Constant Term is the term without a variable.
Let’s see an example of a polynomial that is NOT in descending order, and look at its components. There will not be any like terms in this example.
This function, written in descending order would be.
-2x4 + 5x3 + 3x – 11
The most common mistake made when rewriting a polynomial in descending order is that people misplace the signs of the terms when rearranging. For example, they’d write 2x4 instead of -2x4. The leading coefficient is negative two, not two. So, be cautious here.
The reason descending order is important is that it is organized. As you move through polynomials into quadratics, cubics, and rational functions, you’ll be looking for key features, clues about the equations. These are most easily discovered when the expressions are written in descending order. So, it is best to get in the habit of writing in descending order from the beginning, when things are simpler.
Like Terms must have the same combination of variables and exponents. Let’s see an example and a non-example.
The non-example has two expressions that are not like terms, but they’re close. The combination of variables and exponents is not identical. There is an a to the 3rd power in one, and an a to the 5th power in the other. The exponents for the variable x is different, too.
Combining Like Terms
If you have like terms in a polynomial, you can combine them with addition and subtraction. You can multiply or divide terms that are unlike, but they must be like terms to add or subtract. Terms must be like to add or subtract for the same reason you can’t add apples and oranges. They’re not the same thing. Ultimately, to add or subtract unlike terms is a violation of the order of operations.
Let’s use the expression above to examine how the order of operations dictates that we cannot add unlike terms. First, notice we have two terms, and the second is being subtracted from the first. We do not know the value of either term. In order to take the difference, we must know the values. To find the value we have to square a, then multiply that by 4b. Also, 3 must be multiplied by c’s value, before the value of each term can be subtracted. Since we don’t know the values of a, b, or c, we cannot simplify this expression further.
Addition and Subtraction
Addition is just combining like terms. Let’s see an example of adding two binomials.
Here we are just adding the like terms. There are four total terms, each binomial having two terms. There are only one pair of like terms, however. The -8c and the 5c, are like terms. They combine to make – 3c. The other terms do not share the same exact combination of variables and exponents, so they cannot be combined with addition or subtraction.
Subtraction is a little trickier. Let’s see an example.
Here we are subtracting a positive 3a2b AND negative 8c. Subtracting a negative is the same as addition. Rewriting this expression, we will get the following.
Both terms need to be multiplied by negative one, and negative 8c times negative one is positive 8c.
Once again, we have a pair of like terms, and two terms unlike anything else.
Writing our answer in descending order we get the following.