BEFORE YOU READ THIS POST, PLEASE MAKE SURE YOU HAVE READ PART ONE, WHICH YOU CAN ACCESS HERE.
Hi there! First off, in Part 1, I lied. I said that there would be only 2 parts to the unit on polynomials. Let’s change that to 6-8. 😬
Of course, since this is the last It All Adds Up before my June break, the remaining parts will be added throughout the summer.
When I refer to an exponent during this post, you will see this symbol (^) just before the exponent. For example, two to the second power would be 2^2.
Today, we’ll be focusing solely on multiplying polynomials. To access the lesson, please click CONTINUE READING.
Remember the distributive property? That’s when you distribute a number that is directly next to a parenthesis and distribute it to each monomial in the parenthesis, like in this expression: 6(4x+2). We distribute (multiply) the 6 with the 4x, and then the 6 with the 2. We would end up with 24x +12. I believe I’ve covered different areas of this before, but this same concept can be used throughout mathematics.
As you know, a polynomial is any amount of monomials. To start, we’ll multiply two binomials together. Let’s take these two binomials:
4x + 9 and 7x^2 -72
You can most certainly use the distributive property to multiply both polynomials, though there are two other ways which I will get to later. When multiplying two binomials, you would have to distribute both terms in the first binomial to both terms in the second binomial. If we were to use the polynomials above…
First off, I haven’t spoken about properties of exponents yet, so here is a good link if you’d like to learn about them (they are pretty important): click here.
Actually, it would be very beneficial to you to visit that link, as you will need it when multiplying a variable by a variable.
Let’s start by multiplying 4x by 7x^2, the first step of the distributive property in this particular problem. As you now know, when two of the same variable are multiplied together, you simply add the values of their exponents. Remember: x by itself still has an exponent of 1- we just normally don’t write it that way. So, we multiply the 7 and 4, and the x by the x^2, and reach 28x^3. That’s the first term in the product: keep going and finish distributing.
In the end, you should end up with:
28x^3 + 63x^2 +360
That is the final answer, though it probably isn’t what you ended up with: remember to combine any like terms and write in standard form.
Using A Table
There are, of course, other ways of getting there if the distributive property isn’t your thing. Try using a table!
The graphic above shows how you do so. I’ll give you an example. This is how these two binomials multiplied might look like:
No worries: I changed the font so that you can discern between upper and lowercase. The thing about the distributive property and the table is that it can apply to polynomials of any number of polynomials. Simply distribute appropriately or add the appropriate rows or collumns on the table.
The FOIL Method
In this third way of solving, you need to remember this abbreviation:
Hopefully, based on their colors, you can figure out what each letter refers to using this graphic:
Mind a few parenthesis being colored and awkward spacings, the way you use the foil method is by multiplying the two terms that are considered the first terms, then the outer terms, and so on. You should be left with four monomials, which you can then combine like terms (if there are any) and write in standard form.
I should have mentioned this before, but when two polynomials (any number/s, essentially) are in parenthesis next to each other, we automatically consider them to be multiplied together. The same goes when a variable is placed directly next to a coefficient.
Multiplying Polynomials Other Than Binomials
Like I had said earlier, you can multiply polynomials other than binomials too. You can’t use the foil method here, but you can use tables and the distributive property. With the distributive property, you just have another term to distribute, but you follow the same steps. With a table, add a column for each monomial in the first polynomial and a row for each monomial in the second polynomial.
That’s all folks… until July. Stick around a few more days, though: I have a book review coming on May 29th and several announcement posts coming soon after.
Oh- I’m not creating a Quizlet for this post, but feel free to create your own problems using a variety of exponents and variables.
Thank you so much for reading. Have a great day!