Today's soundtrack is Tim McGraw: Live Like You Were Dying, a solid country rock album. Its title track is still sing-along good. I had the pleasure of seeing Tim McGraw perform with Faith Hill in Vancouver a decade back; it was a great show!
This afternoon, I'm learning about solving polynomial equations.
There are two criteria that must be fulfilled for a multi-term series of numbers to be considered a polynomial:
All exponents must be whole numbers
All coefficients and the constant must be real numbers
There are several ways that we can categorize polynomials.
Standard form
Terms arranged by exponent from highest to lowest
Degree
We call a polynomial a _ degree polynomial, based on the highest exponent. For example, a quadratic polynomial would be called a "second-degree polynomial"
Leading coefficient
The coefficient of the term with the highest variable
Name of polynomial
Again, based on the highest variable
A polynomial with a leading exponent of four is called "quartic," if three, "cubic," if two, "quadratic"
Properties of Polynomial Equations
An nth-dgree polynomial will, if n is an odd number, have a minimum of one root and a maximum of n roots.
An nth-degree polynomial will, if n is an even number, have a minimum of zero roots and a maximum of n roots.
To find roots based on a polynomial's factors, we must remember the Zero Product Property, which tells us that if (A)(B) = 0, then either A or B is zero. We treat all factors as equaling zero; thus, if given the factor (x-2), it is implied that x-2=0; we can move -2 to the right side of the equation to get our root: x=2.
When solving for exact values, we don't break radicals down into decimal form; if they do not perfectly divide into their radical's index, then we leave them as is (but take them down to lowest value whenever possible).
If given a series of roots and asked to find the polynomial equation that they create, just work backwards! By reversing the steps above wherein we convert a factor to a root, we can convert a root to a factor algebraically. After we've done that with each one, it's simply a matter of applying the distributive property to the factors to get the final answer.