Polynomial Operations

1. Polynomial Operations Hanna Jusufi Julia Ly Karl Bryan Ly Huynh Girl : I keep losing stuff when I try to multiply three binomials. Boy : I do, too, but i think crossing off the terms as you rearrange them is a good idea

2. Some Motivation: Super Bass- Nicki Manaj • (For all you Nicki Manaj fans, try to rap this :D) • This one is for the students afraid of polynomials • Top down, you'll know, all these confusing systems • Cause when you come up in the class, you blankin’ out • Got math in class and you're freaking out • You add, subtract, or you might need to divide • All you needa do is don't let it slide • Let’s use, these examples, something you can handle • Polynomials are always here, something you'd wanna strangle **but we'll change that for you

3. Adding Polynomials :). • Add only like terms! • Organize the polynomials in the easiest way you can see it; (vertically/horizontally). • Simplify all terms completely and totally. • Make sure of the signs in front of the parenthesis. • Cancel out any terms if necessary! This is an example of how to add polynomials and a good way to group them to make sure of the signs and like terms! :)

4. Subtracting Polynomials :) • Set the problem up however you are comfortable; (vertically or horizontally). • Remember to completely simplify the polynomials. • Pay attention to the sign infront of the parenthesis and make sure to distribute it to all variables. • Cancel and simplify all variables and terms in the polynomials. This is an example of subtracting horizontally! Notice the distribution of the sign to the variables!

5. Multiplying Polynomials :) • Choose to set up either the polynomials horizontally or vertically; (based on preference). • Keep in mind the signs in front of the variables. • When multiplying the variables that contain an exponent, instead of multiplying them together add them instead. • Always distribute any number, sign, or variable in front of parenthesis. This demonstrates the principle of distributing the terms in front of the parenthesis and making sure of the exponents.

6. ~Dividing Polynomials: Long Division~ • Set up the division where you would normally put the divisor and dividend • To know what to put for the quotient, divide the first terms of the dividend and the divisor and use the product for the quotient ( x^2 / x = x ) Then multiply with the divisor and continue like on any other division • The remainder might not always be zero. With a remainder and a plus sign to the quotient and the remainder would look like a fraction Example: LONG DIVISION Long Division with a Remainder Note: The powers are in descending order

7. Factoring Polynomials What is factoring?Factoring is like the reverse of expanding, therefore it's basically trying to get the polynomial in its standard form that gives you a better visual on seeing the root. Additionally, it's similar to "unsimplifying" a polynomial. To factor, you find Example:

8. Roots • Lets try this out: x^4 - 5x^3 + 3x^2 + 2x + 8You're probably looking at me like you don't know how to do it, but it is in fact, very easy. Because the degree is 4, it is reasonable to use the Rational Root Test. The Rational Root test tells you if the polynomial has a rational zero then it must be a fraction of p/q. P is a factor of the constant and q is a factor of the leading coefficient. The factor of the leading coefficient, 1, is 1, and the factors of the constant term, which is 8, are 1 2 4 8 . Which gives us the possibilities of: ± 1/1 ± 2/1 ± 4/1 ± 8/1. • However, that does not mean these will all work, the ones that work are the ones that are the roots. How do you know? If it equals 0. (You can plug in the numbers for x, use synthetic division, long division, or graph it, to find the zeros. ) As for your answers, you should get:X =2X = 4

9. Reminders When Doing Graphs • Remember: -Polynomials with degrees greater than 1 = curves-Even degrees= shape like U's -Odd degrees = Shape like S's. -The # of the degree is the maximum amount of times that graph can intersect a horizontal axis. • Thered linerepresents how this is apossible graph for an equation with the degree of 3 while thegreen linerepresents how this is a possible graph for an equation with a degree of 4. • ** If you have an idea of how these following graphs look like, you'll beset!

10. A degree of 2 A degree of 3 A degree of 4 A degree of 5 A degree of 6

11. Common Errors/mistake/tips: Polynomial operations exist in any type such as addition, subtraction, multiplication, and division. Students tend to mix all of these operations, however they all have different ways of doing each polynomial so watch out for your signs! When you see something like (x+4)+(x-9) you add just like that. However, that is completely different from something like (x+4)(x-9). The first one is adding straight across. The 2nd one is multiplying so you must distribute the x. Kind of like foiling. DON'T GET THEM MIXED UP! I WARNED YOU. Recheck your work. If you find a root that equals 0, don't be tricked. It may be an extraneous answer so plug it in your equation! And remember, all polynomial equations can not always factor out so if you're 100% sure that there is a polynomial that you cannot factor out, the answer to that polynomial will be "No solution" If you are multiplying two or more equations together make sure that the equation you have ended up with is in the order of descending exponents. For example if you are evaluating if you get the equation x^2-5x+2-x^3 this is wrong and you would need to put it in the order of x^3+x^2-5x+2.

12. For all you One Direction Fans:And even if you aren't, it's catchy. What Makes You Beautiful- One Direction You're insecure, don't know what for, Polynomial Operations leave you wanting mo-o-re, Whether to subtract, or maybe add, Multiply, and or divi-i-de, Everyone else in the class can do it, Everyone including you! Baby it ain't hard to complete these operations, The way that you make sure your accurate gets me overwhelmed, But when you check yours signs and distribute, It ain't hard to tell, You don't know, Oh, Oh, You don't know you are super sma-ar-art!