Remainder & Factor Theorems

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Author:
Gymnastxoxo17
ID:
267144
Filename:
Remainder & Factor Theorems
Updated:
2014-06-18 06:45:04
Tags:
Chapter Four
Folders:
H Pre-Calc
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d
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  1. 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
  2. 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
  3. 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
  4. the maximum number of real zeros is equal to ?
    the degree of the polynomial
  5. 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
  6. depressed equation
    resulting quotient after you divide by the zero
  7. how to solve a polynomial equation
    find the zeros of it; these are the solutions
  8. irreducible
    a quadratic factor ax2+bx+c that cannot be factored over the real #s; you cannot factor it and get real #s
  9. a polynomial (with real coefficients) of odd degree has how many real zeros?
    at least one
  10. real number
    any number in the number system with the regular number line
  11. imaginary number
    things like i
  12. 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
  13. complex polynomial
    polynomial where all of the coefficients are complex numbers and the exponents are nonnegative integers and the coefficients are complex numbers
  14. fundamental theorem of algebra
    every complex polynomial of at least one degree  has at least one complex zero
  15. every complex polynomial of at least one degree has ? complex zeros
    exactly n, and they can repeat
  16. conjugate pairs theorem
    take a polynomial with real number coefficients. If r=a+bi is a zero of f, the complex conjugate a-bi is also a zero of f
  17. a polynomial with real coefficients of  odd degree has ? real zeroes
    at least one
  18. i2
    -1

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