The connection between factors and zeros is pretty simple, the zeros that you find from an equation is the factored function of that same equation. For example in the equation above, the zeros that were found are the numbers used in the factored form of the polynomial. Division helps us factor polynomials because it is a quicker way to find the solutions. The degree of the polynomial helps to predict the number of zeros by telling us in the highest degree, in the polynomial above the highest degree is 4, therefore that polynomial should have 4 zeros. Even though it does not look like it this polynomial has 4 zeros. There is one repeating zero in this equation which you would find out when using division (another reason division helps us factor polynomials). The highest power of the degree may not always tell us the number of factors because, like the example, there could be a repeated zero. Also it may not give the number of factors because it could be something that gives you an "ugly" square root or an imaginary solution.
Even and odd functions are similar in some ways. For example, both even and odd functions have patterns and trends in their equations. They each have a certain way their equations need to be set up in order to be even or odd. Even and odd functions are different because even function graphs are symmetrical, they will always be symmetrical and if they aren't then they are not even functions. Not only will an even functions graph be symmetrical but their exponent in their equation will be an even number. However, in an odd function the exponent will always be an odd number. If you are checking to see if an equation is even you will need to replace the x with a negative x, if you get a positive x in the end then you've got yourself an even function. To check and see if it's an odd function you will need to replace the x with a negative x, just like even functions, but this time you want to get a negative x. For example, an even function that looks like f(x)=x^4 will want to end up looking like f(x)=x^4. You want to solve algebraically and get the exact function you started with. An odd function that looks like f(x)=x^3-x will want to end up looking like f(x)=-x^3+x, so you want to get the opposite of the equation you started with. Some families of functions that will always be even are quadratics, square roots, and some others. These families will be even as long as their "h" always equals zero. Families that will always be odd would be sine functions and some linear functions. These families will be odd if their inflection point is on the x axis.
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