Adding and Subtracting Polynomials

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 go over how to add and subtract polynomials. The slides progress through explaining what function notation is as well as its uses. Then we discuss how to put a polynomial in standard form and to combine it with another polynomial to with addition or subtraction.

So without further ado, read through the slides below to get a feel for how to add and subtract polynomials!

Add 1Add 2Add 3Add 4Add 5Add 6Add 7Add 8Add 9Add 10Add 11Add 12Add 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!

Multiplying Polynomials

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

  • Polynomials
  • Adding and Subtracting Polynomials
  • Objectives
  • At the end of this lesson you should feel comfortable:
  • Recognizing function notation
  • Adding polynomials
  • Subtracting polynomials
  • Operations with polynomials
  • Until now we have only looked at polynomial expressions, so naturally we need to make things a little more complicated.
  • Introducing function notation:
  • Ugly, ugly, ugly, but necessary to give the general form of a polynomial
  • This is just an equation with f(x)=blah instead of y=blah.
  • It is useful to show what you are plugging into the function such as f(1), f(-2), f(x-3)
  • Operations with Polynomials
  • f(x) is just a way of saying this is a function where x is the input.
  • (or you can just think of f(x) as y)
  • The a’s represent coefficients which are the numbers in front of the x’s
  • Notice that the little numbers beside them called subscripts make each distinct
  • The exponents are labeled as n, n-1, n-2 etc because our polynomial must be in descending order (decreasing exponents)
  • Operations with polynomials
  • Ok, I hear you. It’s still not something you want to think about, so how about an example to make it concrete?
  • Better? Same thing but with real numbers to play with.
  • Our function is still in descending order where my exponents get smaller and smaller
  • (This is also called a cubic polynomial of 4 terms in case you’re wondering.)
  • Operations with polynomials
  • So let’s say we’ve got a couple of functions, f(x) and g(x).
  • Stands to reason we can do some things with them like we would with numbers like so:
  • Phew, that’s a lot to take in, so for now let’s only focus on the first two, adding and subtracting polynomial functions
  • Adding polynomials
  • Let’s take a look at two functions.
  • Here we can see f(x) is a quadratic trinomial and g(x) is a cubic polynomial of 4 terms.
  • Lets say we want to add f and g, denoted like so:
  • Then this means:
  • Adding polynomials
  • Notice that we’ve left both functions in parentheses to avoid sign errors
  • (This will be especially important when subtracting polynomials)
  • Next, we should rearrange our terms so that like terms are next to one another
  • Begin with the largest exponent
  • Then work our way down
  • This way our answer will be in standard form, which means the polynomial is in descending order
  • Adding polynomials
  • Now we are back to simply combining like terms!
  • We’ve only got one x cubed term, so it stays the same
  • 5 and 1 make 6 x squared terms
  • -2 and 3 make 1 x term
  • 3 and -4 make -1 as the constant
  • And viola! We have completed our addition!
  • Subtracting polynomials
  • Now let’s take our same two functions, f and g, and subtract them.
  • Notice the parentheses still in place here around both functions
  • This is especially important in order to keep track of the signs in our second function after we subtract each term.
  • Because we are subtracting the whole thing, we change the signs of each term.
  • Subtracting polynomials
  • And now, yet again, back to what we know, combining like terms!
  • We’ve only got one x cubed term
  • 5 minus 1 is 4 x squared terms
  • -2 minus 3 is -5 x terms
  • And lastly, 3 plus 4 is 7 as our constant
  • Huzzah, another completed polynomial operation!
  • Which way?
  • Notice that you may add two polynomials in any order
  • f(x)+g(x)=g(x)+f(x)
  • This is called the commutative property
  • 3+4=4+3 because both are equal to 7
  • Polynomials operate the same as adding because adding polynomials is the same as adding each like term
  • Which way?
  • Subtraction, on the other hand, is not commutative
  • f(x)-g(x) does not equal g(x)-f(x)
  • Notice what happens after we distribute the negative through in both cases.
  • The signs are opposite for each of the terms, which will add up to different equations after combining like terms.
  • conclusion
  • Function notation is used to name a function and show what the input is with f(x).
  • We may perform operations on polynomials much like we do with real numbers.
  • Some polynomial operations are commutative, meaning they can be done in either order
  • Subtracting polynomials requires us to first change the signs of the second polynomial before combining terms.
  • Adding polynomials simply requires us to combine like terms.