Multiplying 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 multiply polynomials by working piece by piece. The slides discuss how to multiply the coefficients first and then add the exponents on the variables. Next the lesson discusses how to distribute and builds off this concept to introduce multiplying binomials by using the FOIL method.

So without further ado, read through the slides below to get a feel for how to multiply polynomials together!Mult 1Mult 2Mult 3Mult 4Mult 5Mult 6Mult 7Mult 8Mult 9

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!

Coming Soon…

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

  • Polynomials
  • Multiplying Polynomials
  • objectives
  • By the end of this lesson you should feel comfortable:
  • Multiplying terms
  • Using the distributive property
  • Multiplying binomials
  • Multiplying polynomials of larger terms
  • Multiplying polynomials
  • Let’s start with two monomials
  • (That means polynomials with one term, remember?)
  • The trick to multiplying terms is to take it one piece at a time.
  • First, let’s deal with the numbers in front called coefficients. 5 times -2 is -10.
  • For our variables since we are multiplying with the same base, we can add our exponents. 2+3=5
  • Multiplying polynomials
  • So now that we know how to multiply terms, how about we take another step?
  • Remember the distributive property?
  • The 4 on the outside gets distributed to both the pieces inside the parentheses.
  • 4 times 3x is 12x
  • 4 times 7 is 28 (with the plus still in front as it remains positive)
  • Multiplying polynomials
  • Now we know that we can multiply terms, so let’s apply the distributive property to some expressions!
  • Looks eerily similar doesn’t it?
  • The process is the exact same, as well.
  • First we multiply 4x and 3x, which is 12 x squared
  • Then we multiply 4x and 7 to get 28x
  • Congratulations, you just distributed!
  • Multiplying polynomials
  • And we can take things even further!
  • What if we had two binomials?
  • The trick is to distribute twice, once for each term.
  • I’ll show you the way I like to arrange the products to easily combine like terms
  • Start with both our First terms.
  • Next the two Outside terms.
  • Then the two Inside terms.
  • Finally the Last two terms.
  • Multiplying polynomials
  • Start with both our First terms.
  • Next the two Outside terms.
  • Then the two Inside terms.
  • Finally the Last two terms.
  • This is called the FOIL method for the first letter of each step.
  • Then we just have to combine like terms:
  • Notice how we lined up like terms so that we can see exactly what can be combined.
  • Multiplying Polynomials
  • And yes, there is even one more step beyond multiplying binomials…
  • Larger polynomials!
  • We follow the same pattern of distributing, but with more iterations:
  • Multiply the first term by each of the terms in the second polynomial
  • Then the second
  • Then the third
  • Multiplying polynomials
  • The last step is to combine like terms, as always.
  • We’ve only got one x fourth term
  • -12 plus 4 gives us -8 x cubed terms
  • 24 plus -8 plus -10 gives us 6 x squared terms
  • 16 plus 20 gives us 36 x terms
  • Finally, -40 is just hanging out by itself
  • Tada! Now you can multiply any two polynomials together!
  • Just follow the pattern of multiplying term by term!
  • conclusion
  • To multiply terms you must go piece by piece.
  • First multiply the numbers, coefficients.
  • Then multiply the variables by adding the exponents.
  • The distributive property is used to multiply a monomial by each term in a larger polynomial.
  • Multiplying binomials requires distributing both terms in the first to both terms in the second.
  • Multiplying larger polynomials follows the pattern of multiplying binomials by distributing each term in the first polynomial to each term in the second.
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