Polynomials

Unit 6 - Polynomials
Big ideas: robert w:
 * [|Essential Topics: Polynomials]
 * a line is ^1, a bounce is ^2 (or any even power), and a wiggle is ^3 (or any odd power)
 * || even exponent || odd exponent ||
 * positive LC || up up || down up ||
 * negative LC || down down || up down ||


 * ^^^^^^^^^^^^^^^^^^^^^^ this is the end behavior chart assuming reading the graph from left to right! (its easier to just think about it though!)
 * if you add the exponents and the sum is even, the end behaviors are the same. if you add the exponents and the sum is odd, the end behaviors are opposite.
 * if you add the exponents and the sum is even, the end behaviors are the same. if you add the exponents and the sum is odd, the end behaviors are opposite.

Why does a positive leading coefficient always mean the graph goes up up up down.... same with the negative LC... The leading coefficient tells what the right side does so if it is positive the right side is up and if it is negative the right side is down. However the sum of the //exponents// tells what the left side does compared to the right side (see the bullet above)

Helpful hints: Steps To graphing Exponential (another way to explain: With Nigel Walker)

Decarte's rule of signs Sample Problems: (click here for solutions to these problems)
 * the form is of an exponential is y = kx then:
 * if //k// is positive then the right side of the graph is up; if //k//is negative, the right side of the graph is negative.
 * the exponent tells us how graph will touch the x-axis (pass through, bounce, "wiggle")
 * If the exponent is 1 then the graph will pass through the x-axis
 * [[image:Screen001.jpg]]
 * If the exponent is even then the graph will bounce off of the x-axis.
 * [[image:Screen002.jpg]]
 * If the exponent is greater than or equal to 3 and odd, then the graph will "wiggle"
 * [[image:Screen003.jpg]]
 * REMEMBER if an imaginary term is part of a factored polynomial, then the conjugate is also a factor **(assuming the polynomial has rational coefficients) robert w**
 * ie. if (x + (2+3i)) is a factor, than (x + (2-3i)) is also a factor
 * to get rid of the "i" easily, shift the parenthesis to make the terms a difference of squares!
 * ** (x + (2+3i)) * (x - (2-3i)) ** IS EQUAL TO ** ((x+2) + 3i) * ((x + 2) - 3i) **which simplifies to x^2 - 4x + 4 + **__9 <--- don't forget to square the 3 as well as eliminate the "i"__**
 * this whole idea is the same for irrational roots. (see page 314 in the book)---> that page really helps explain that and imaginary roots
 * if a polynomial **P(x)** is divided by **(x - r)** then the remainder of the division is the same as solving for **P(r)** !!!
 * long division works for EVERYTHING!
 * synthetic division only works when you have a **linear denominator**if your dealing with a fraction
 * to get all the factors, use the rational root theorem
 * rational root theorem means take the (+) and (-) values of all factors of the highest degree term and the last term (the terms without a variable)
 * rational root theorem means take the (+) and (-) values of all factors of the highest degree term and the last term (the terms without a variable)
 * for example if you had y = x^4 - x^3 + x^2 - x^1 - 1 you would count the sign changes so in this example there are 2 (Aren't there //**3**// sign changes?)
 * then, plug in a - and count + + + + - so there is only 1 negative
 * set up a table of all possible signs with +, -, and complex (complex roots come in pairs); since the highest exponent is 4, there can be a max 4 roots
 * + - complex
 * 2 0 2
 * 1 1 2
 * 0 0 4
 * i think this is right, if its not then correct it
 * 1) factor the equation: x^3 + 3x^2 - 25x - 75 robert w
 * 2) List all possible rational roots of the following equation: 3x^3-2x^3+7x^2-5 -Bailey
 * 3) What is the leading coefficient and the degree of this equation: 87x^4+23x^2-9x+333 -- Priyanka
 * 4) So this is kind of a big overall question...but here goes: Graph on one graph. y=1/((x-2)(x+3)^2(x-5)^5), x^2-36x+ - 4y^2-9y=100, 4y^2=16 + 25x, y=(x+3)^2(x-1)^5(x+7)^3
 * 5) factor and graph the equation 3x^3 + 12x^2 - 3x - 12 --taylor b.
 * 6) Graph: -8 (x - 5)^5 + 4
 * 7) If (x+2) is a factor of x^4-2x^3+bx^2-bx+10, then what is b? -Emily
 * 8) Find the solutions for 2x^3-2x^2-4x=0 Mallory
 * 9) graph: x^4 - 2x^3 - 7x^2 + 8x + 12 robert w
 * 10) graph: -x^5 + 10x^4 - 24x^3 - 10x^2 + 25x robert w