Remainder & Factor Theorems
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remainder theorem
if f(x) is divided by a factor, the remainder of doing this will be the zero of that factor plugged into the function and solved for

factor theorem
 if something is a factor of a polynomial, its remainder using the remainder theorem will be zero
 also, if something is a factor of a polynomial, then if you plug its zero in and solve you will get no remainder

rational zeros theorem
 when you have a polynomial of at least one degree with integer coefficients, you can make a list of all the potential zeros by dividing p/q
 p is all the integer factors of the constant
 q is all the integer factors of the leading coefficient

the maximum number of real zeros is equal to ?
the degree of the polynomial

how to find the real zeros of a polynomial
 the max number of real zeros is equal to the degree
 use the rational zeros theorem to identify rational #s that are potential zeros
 use a calc to make a smart choice about plausible zeros to test
 use the factor theorem to see if you're right, then use division to factor the polynomial
 repeat until the polynomial cannot be factored out anymore
 use zero product property to find the zeros

depressed equation
resulting quotient after you divide by the zero

how to solve a polynomial equation
find the zeros of it; these are the solutions

irreducible
a quadratic factor ax^{2}+bx+c that cannot be factored over the real #s; you cannot factor it and get real #s

a polynomial (with real coefficients) of odd degree has how many real zeros?
at least one

real number
any number in the number system with the regular number line

imaginary number
things like i

complex numbers
 numbers that are a combination of real and imaginary numbers, like a+bi
 any part can be zero, so an imaginary number or real number alone are considered complex numbers too

complex polynomial
polynomial where all of the coefficients are complex numbers and the exponents are nonnegative integers and the coefficients are complex numbers

fundamental theorem of algebra
every complex polynomial of at least one degree has at least one complex zero

every complex polynomial of at least one degree has ? complex zeros
exactly n, and they can repeat

conjugate pairs theorem
take a polynomial with real number coefficients. If r=a+bi is a zero of f, the complex conjugate abi is also a zero of f

a polynomial with real coefficients of odd degree has ? real zeroes
at least one
