2. Basics<br />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.<br /> <br />Remainder Factor: If a polynomial f(x) is divided by (x-k) then the remainder is r=f(k)<br />Factor Theorem: A polynomial f(x) has a factor (x-k) if and only if f(k)=0.<br />
3. Detailed<br />Standard form: ax²+bx+c<br /> The whole problem must be multiplied out. There can be no parenthesis left.<br />Factored form: a(x+r₁)(x+r₂)<br />Degree: The highest power of X when in standard form<br />Leading co-efficient: first co-efficient when the polynomial is in standard form and in order (high-low)<br />
4. Graphing Functions<br />Root behavior tells what happens in between the ends of graphed function. <br />A pass through anything to the first power.<br />A bounce off is anything to an even power.<br />A squiggle through is anything to an odd power. <br />
5. Graphing Functions Continued …<br />End behavior is what happens at the end of a graphed function.<br />A graph that might help determine how to graph a polynomial function is:<br /> Even Odd <br />Normal: ↑↑ ↓↑<br />Negative: ↓↓ ↑↓<br />
6. Dividing Polynomials<br />Long Division: the same way as done in elementary school.<br />Synthetic Division: a shortcut used for most problems. Synthetic division can only be used if the divisor is not to any power. <br />Ex: (4x⁴+2x³-7x+1) / (x-4) can be solved by synthetic division.<br /> (4x⁴+2x³-7x+1) / (x²-4) can’t be solved by synthetic division because the divisor is squared.<br />
7. Write down first part of <br /> equation all over second <br /> part of equation. Then do <br /> normal division as you <br /> normally would. Make sure <br /> you multiply everything <br /> right and subtract right. <br />The first subtraction you make the first part of it should<br /> equal zero and don’t forget to carry down your next <br />number. <br />Long Division<br />
8. Synthetic Division<br />Obviously this problem would have been (-1x³+8x²+63) / (x-4)<br />How to do synthetic division:<br /> 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)<br />
9. Remainder Theorem<br />If a polynomial f(x) is divided by (x-k) then the remainder is r=(f(k).<br />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.<br />
10. Factor Theorem<br />A polynomial f(x) has a factor (x-k) if and only if f(k)=0.<br />If you get a remainder of zero you have found two factors of the polynomial.<br />
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