Tips+and+Tricks

**Tips and Tricks** (from the sheets)
1. __Properties of Polynomial Functions & Determining Equations__ - odd degrees mean an even number of turning points - even degrees mean that both end behaviours are in the same direction - when f(x) is an actual number, sub into the equation to find A eg. f(-2) = 90 y = a (x-1) (x-3) (x+5) 90 = a ( -2 -1) ( -2 -3) ( -2 +5) - read all instructions - if an odd function has an odd degree then it must pass through (0,0)

2. __Dividing Polynomials__ - if one polynomial isn't there, add it with a 0 as the coefficient eg. 6x^4 - 3x^2 --> 6x^4 + 0x^3 - 3x^2 - its easier to work with fractions in long division as opposed to synthetic division - don't forget negatives, signs are important - synthetic is //adding//, long division is //subtracting// - if leading coefficient is 1 then use synthetic - if leading coefficient is >1 then use long division - in long division, double-check signs

3. __Factoring & Solving Polynomials__ - move everything to one side - add 0 if a term is missing - take your time - watch out for signs - make sure you know difference/sum of cubes

4. __Graphing Polynomials__ - find midpoint between roots to approximate turning point - find y-int (when x=0) - double roots only //touch// axis, they don't cross - positive or negative coefficients will determine end behaviour - count the number of x's in factored form to determine the degree

5. __Polynomial Inequalities__ - put the expression in the form f(x) > 0 or f(x) < 0 etc (move everything to one side) - factor out any common factors - find the values of x where y=0 - break up the domain into intervals and find where your expression is true using a) chart b) graph or c) number line - watch out for points touching the x-axis if f(x) is less than and equal to 0 or more than and equal to 0

__Difference/Sum of Cubes__ Remember these:
 * //a//3 + //b//3 = (//a// + //b//)(//a//2 – //ab// + //b//2) **
 * //a//3 – //b//3 = (//a// – //b//)(//a//2 + //ab// + //b//2) **

- Mnemonic for the signs: **SOAP** "same" as the sign in the middle of the original expression, "opposite" sign, and "always positive" - __ Note: __ quadratic part does not factor completely []
 * //a//3 ± //b//3 = (//a// [//same sign//] //b//)(//a//2 [//opposite sign//] //ab// [//always// //positive//] //b//2) **