Basics In order for the problem to be a polynomial function the exponents must be a positive and whole number. You must know what standard form, factored form, degrees, and leading co-efficient mean. (See first entry with power point) You must know what root and end behaviors are as well and how to graph and/or read them. (See second entry) You must know how to divide polynomials using long division and synthetic division, and lastly you must know what the remainder and factor theorem mean.
Remainder Factor: If a polynomial f(x) is divided by (x-k) then the remainder is r=f(k) Factor Theorem: A polynomial f(x) has a factor (x-k) if and only if f(k)=0.
Detailed Standard form: ax²+bx+c The whole problem must be multiplied out. There can be no parenthesis left. Factored form: a(x+r₁)(x+r₂) Degree: The highest power of X when in standard form Leading co-efficient: first co-efficient when the polynomial is in standard form and in order (high-low)
Graphing Functions Root behavior tells what happens in between the ends of graphed function. A pass through anything to the first power. A bounce off is anything to an even power. A squiggle through is anything to an odd power.
Graphing Functions Continued … End behavior is what happens at the end of a graphed function. A graph that might help determine how to graph a polynomial function is: Even Odd Normal: ↑↑ ↓↑ Negative: ↓↓ ↑↓
Dividing Polynomials Long Division: the same way as done in elementary school. Synthetic Division: a shortcut used for most problems. Synthetic division can only be used if the divisor is not to any power. Ex: (4x⁴+2x³-7x+1) / (x-4) can be solved by synthetic division. (4x⁴+2x³-7x+1) / (x²-4) can’t be solved by synthetic division because the divisor is squared.
Write down first part of equation all over second part of equation. Then do normal division as you normally would. Make sure you multiply everything right and subtract right. The first subtraction you make the first part of it should equal zero and don’t forget to carry down your next number. Long Division
Synthetic Division Obviously this problem would have been (-1x³+8x²+63) / (x-4) How to do synthetic division: Carry the first number down (-1) and multiply by the divisor (4), then write that under the next number (8). Add the two numbers together (8-4=4) Write the 4 under the line then multiply 4 by the divisor. Add that to the next number. (0+16=16) multiply 16 by divisor and write under next number and add. (63+64=127)
Remainder Theorem If a polynomial f(x) is divided by (x-k) then the remainder is r=(f(k). Basically if a problem tells you to use the remainder theorem all they want is the remainder so if synthetic division is available to use it’s a lot quicker.
Factor Theorem A polynomial f(x) has a factor (x-k) if and only if f(k)=0. If you get a remainder of zero you have found two factors of the polynomial.