In Algebra, Polynomials are commonly used as lessons, and you might see them more than once. I have been learning all about polynomials for awhile now, and in my eyes, they are easy and fun. However, this may not be the case for everyone, so I want to give everyone a chance to have fun with polynomials and excel in them. I love polynomials, and I hope you do too.
Types of Polynomials
There are many polynomials, and here they are!
Binomial- 2 terms
Trinomial- 3 terms
Mono- is the Latin root for 1
Bi- is the Latin root for 2
Tri- is the Latin root for 3
5x is a monomial
2x+5 is a binomial
x squared+x+5 is a trinomial
I can't do the small 2 for squared on this lens, so I just write out the word squared instead.
One of the first steps to polynomials is learning how to add and subtract them. You can only add and subtract like terms. Let's take a look at the first example.
This is a tricky one. 2x squared and 5x squared definitely add up to 7 x squared. However, you must also know that you cannot add -5x and 8 together because they are not like terms. You can add 17 and 8 together because they are like terms. If you have a problem remembering this, you can put 0x under -5x (0x can be plus or minus because it does not affect the equation, it's just a place holder). Then the equation would look like this:
Was that easier? You have to see that sometimes, a polynomial will be out of place. Your teachers will try to trick newbies with this method so that they can learn from their mistakes. That is why the placeholder trick can help out.
Subtracting Polynomials can be converted to addition, which would make things a lot easier. Subtracting Polynomials is like adding their opposites, and that is the best approach to solving them. Let's look at one example.
You can change the minus sign to a plus sign, and then switch everything else. Remember that 3x squared is positive, but the minus sign is in front. This is what the new equation would look like. THE TOP REMAINS THE SAME
+ -3x squared-7x+8
Now, isn't that easier? 2x squared plus negative 3x squared is negative x squared. 6x plus a negative 7x equals -x, and -8+8=0.
This is the final answer.
+ -3x squared-7x+8
Negative x squared minus x plus 0.
For beginners, I recommend adding the 0 so that you know it's there, but as you learn about polynomials for a long time, you should take out the 0.
Multiplying Polynomials Part 1
FOIL is the term that is used to help out when we have to multiply two binomials.
You can think of FOIL as a Punnett Square, and the picture perfectly describes one way that you can approach FOIL.
Let's look at the example in the Punnett Square.
This is what it would look like:
(x times x)+(x times 6)+(3 times x)+(3 times 6)
F O I L
Your final answer is this:
TIP: When the value of both x's is 1, you will always get x squared, and if you add the two constants together (numbers without variables such as 3 and 6) and get the x value. So, in (x+6) (x+3), just do 3+6 and put the x at the end of the number. That is how we got 9x for the x term. You cannot use this method if x>1.
So, in (2x+5) (3x+6), you will have to do some more math to find the answer.
So, with the method of FOIL, the answer was.....
6x squared+27x+30. FOIL will help you solve the multiplication of two binomials.
Squaring a Binomial
If you EVER see (x+1) squared, then it means (x+1) (x+1). Writing it out will prevent you from forgetting the middle term which in this sample would be 2x.
Dividing A Trinomial
When you divide a trinomial by a binomial, 99% of the time, that binomial is a factor of the trinomial. Let's take a look at an example:
The 2x+1 is a factor of the trinomial.
Now, we have to find out what the ??? stands for. The two constants must have a product of 2, and you must have a middle term of 5x. Also, you must have 2x squared. Since you already have 2x, all you need now is x. Then, through testing out the factors of 2, you will realize that 2 is the constant. So, now the equation looks like this:
The 2x+1's cancel out leaving you with (x+2). You can check the binomials first through FOIL, but they will bring you back to 2x squared+5x+2.
Awesome Books About Polynomials
Look Out For This One!
There is one thing that you must be aware of when you are solving these kinds of polynomials. Let's look at the example below.
The most common mistake most people make is squaring what is in the parenthesis to get an answer of x squared+4. This is incorrect because you multiply the polynomial by the same polynomial. Look at the example below.
So, now you know what it means to square the number. It may look the same as x squared+4, but when you write out the entire polynomial, you will realize that there IS a middle term. Look out for this one.
Bugs Bunny isn't Daffy Duck and Daffy Duck isn't Bugs Bunny.