Question 534459: List all possible rational zeros for the polynomial below. Find all real zeros of the polynomial below and factor completely. Please show all of your work.
f(x) = 3x^4+17x^3+17x^2-33x-36
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! List all possible rational zeros for the polynomial below. Find all real zeros of the polynomial below and factor completely. Please show all of your work.
f(x) = 3x^4+17x^3+17x^2-33x-36
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Use Rational Roots Theorem to find real zeros:
....0...|......3........17.......17........-33.......-36..............
....1...|......3........20.......37..........4.........-32.......
....2...|......3........23.......63.........93.........150 (2 is upper limit), (root between 1 & 2)
.........................................................
....0...|......3........17.......17........-33.......-36
..-1...|......3........14........3.........-36........ 0 (-1 is a root)
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....0...|......3........14........3.........-36
....1...|......3........17.......20........-16...................
....2...|......3........20.......43.........50 (2 is upper limit), (root between 1 & 2)
.........................................................
....0...|......3........14........3.........-36
..-1...|......3........11......-8.........-28..................
..-2...|......3.........8.......-13.......-10....................
..-3...|......3.........5.......-12..........0 (-3 is a root)
f(x)=(x+1)(x+3)(3x^2+5x-12)
f(x)=(x+1)(x+3)(3x-4)(x+3)
Zeros are:-3(multiplicity 2), -1, and 4/3
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Possible Rational Roots: p=factors of 36 (constant), q=factors of 3(lead coefficient)
...q\p...|..±1........±2.......±3.....±4......±6......±9.....±12.......±18.........±36.
...±1....|..±1.........±2......±3.....±4......±6......±9.....±12........±18.........±36
...±3....|..±1/3....±2/3...........±4/3.......................................................±12
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