Polynomials and Terms

The lesson below is still images taken from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. If the PowerPoint is not one of my freebies, you may also head over to my YouTube channel to see the slideshow fully animated and can pause as needed to be sure you grasp each concept before moving forward (each lesson will always be free there!).

In this lesson we discover what Polynomials are. The slides introduce these expressions by breaking down the word and discussing how to identify terms. The lesson then goes on to combining like terms in order to simplify longer and more complicated expressions so they may be written in standard form. Then we are able to name polynomials based on the degree and total number of terms.

So without further ado, read through the slides below to get a feel for what polynomials are and how to simplify and name them!

Poly 1Poly 2Poly 3Poly 4Poly 5Poly 6Poly 7Poly 8Poly 9Poly 10Poly 11Poly 12Poly 13

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

Adding and Subtracting Polynomials

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

  • Polynomials
  • An introduction
  • objectives
  • By the end of this lesson you should feel comfortable:
  • Recognizing polynomials
  • Identifying and counting terms
  • Combining like terms
  • And naming polynomials
  • What are polynomials?
  • Break the word down into parts:
  • Poly-meaning many
  • Nomial-having to do with numbers or terms
  • A polynomial is an expression with many terms
  • Ex
  • What are terms?
  • Terms are parts of a polynomial separated by +’s and –’s
  • Like Terms
  • Like Terms-terms that have the same variable or variables raised to the same powers
  • Ex
  • Other terms that have different exponents or variables are not alike
  • Ex
  • Combining like terms
  • So what’s the point?
  • MAKING THINGS SIMPLER!
  • This is an ugly polynomial with six terms.
  • More terms, means more chances to make mistakes.
  • So we clean it up.
  • Combining like terms
  • Combining like terms
  • All well and good, but why? Good question!
  • Mathematicians don’t move stuff around without reason. We are too lazy for that.
  • Now we can add our like terms.
  • We have 3 somethings and another something first in the red boxes. It just so happens that the somethings are “x squared’s”
  • So 3 and 1 together make 4 somethings or:
  • Combining like terms
  • Then we can follow suit with the other like terms.
  • Add up our -5 and 2 x’s:
  • Add up our 2 and -4 as contants:
  • Then viola! A much simpler looking ordeal than we started with.
  • Naming polynomials
  • Now that we have our polynomial cleaned up, we can properly name it.
  • Names require two pieces of information:
  • The Degree
  • The Number of terms
  • The Degree is the highest exponent (or the exponent in the leading term)
  • In our case, we can see that our degree is 2
  • Naming polynomials
  • We state the degree by saying either nth degree, where n is the number of the exponent, or following along the naming convention:
  • 1 – Linear
  • 2 – Quadratic
  • 3 – Cubic
  • 4 – Quartic
  • Then using 5th degree, 6th degree and so on…
  • But that’s not all!
  • Naming polynomials
  • We also said that there is another important part to naming polynomials, the number of terms.
  • Earlier, we discussed what terms are, so you should be able to identify the three terms here.
  • The naming convention for terms is a prefix followed by –nomial.
  • Monomial
  • Binomial
  • Trinomial
  • Then a polynomial of 4 terms, 5 terms, etc…
  • Naming polynomials
  • Therefore with a degree of 2, we have a quadratic
  • And with 3 terms we have a trinomial
  • Thus the name of our polynomial is a
  • Quadratic Trinomial
  • Conclusion
  • Polynomials are expressions composed of many terms that have different exponents.
  • Terms are smaller pieces of an expression separated by +’s and –’s.
  • Terms are considered alike if they have the same variable and the same exponent
  • You may combine like terms to simplify your polynomial
  • Standard form is where you list terms in descending order by exponent
  • The degree of a polynomial is the highest exponent.
  • Naming polynomials requires you to know the degree and number of terms