SOLUTION: 9x^4-18x^3+17x^2-18x+8 Unsure how to complete this. How to find the zero(rational) roots

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Question 1000536: 9x^4-18x^3+17x^2-18x+8
Unsure how to complete this. How to find the zero(rational) roots
Found 3 solutions by stanbon, MathLover1, solver91311:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
9x^4-18x^3+17x^2-18x+8 = 0
Unsure how to complete this. How to find the zero(rational) roots
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I graphed it and found roots at x = 2/3 and x = 4/3
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Use synthetic division to find other roots
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Cheers,
Stan H.
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Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

9x%5E4-18x%5E3%2B17x%5E2-18x%2B8=0.......factor completely,..first write -18x%5E3 as -6x%5E3-6x-12x%5E3, 17x%5E2 as 9x%5E2%2B8x%5E2, and -18x as -6x-12x
9x%5E4-6x%5E3-6x-12x%5E3%2B9x%5E2%2B8x%5E2-6x-12x%2B8=0......group
%289x%5E4%2B+9x%5E2%29-%2812x%5E3%2B12x%29-%286x%5E3%2B6x%29%2B%288x%5E2%2B8%29=0
9x%5E2%28x%5E2%2B+1%29-12x%28x%5E2%2B1%29-6x%28x%5E2%2B1%29%2B8%28x%5E2%2B1%29=0
%289x%5E2-6x-12x%2B8%29%28x%5E2%2B1%29+=+0
%28%289x%5E2-6x%29-%2812x-8%29%29%28x%5E2%2B1%29+=+0
%283x%283x-2%29-4%283x-2%29%29%28x%5E2%2B1%29+=+0
%283x-4%29+%283x-2%29+%28x%5E2%2B1%29+=+0
solutions:
if %283x-4%29++=+0=>3x=4=>highlight%28x=4%2F3%29
if %283x-2%29++=+0=>3x=2=>highlight%28x=2%2F3%29
if+%28x%5E2%2B1%29+=+0=>x%5E2=-1=>x=sqrt%28-1%29=>highlight%28x=i%29 and highlight%28x=-i%29

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Use the Rational Roots Test. If a polynomial function has a rational zero, it will be of the form where is a factor of the constant term and is a factor of the lead coefficient.

Using the Rational Roots Test, create your list of possible zeros. Counting both positive and negative possibilities you have 24 possible rational zeros for your polynomial. The other two zeros are complex.

Then start testing them using Synthetic Division. I'll save you a little work: once you have found two real zeros, you have found them all. The other two zeros are complex.

If you need a refresher on Synthetic Division, look here:

Purple Math Synthetic Division

Note that there are 4 pages; read them all.

John

My calculator said it, I believe it, that settles it