SOLUTION: Use Descartes' Rule of Sign to determine how many positive and negative zeros each polynomial function may have. {{{f(x) = -6x^5 + x^4 + 5x^3 + x + 1}}}

Algebra ->  Trigonometry-basics -> SOLUTION: Use Descartes' Rule of Sign to determine how many positive and negative zeros each polynomial function may have. {{{f(x) = -6x^5 + x^4 + 5x^3 + x + 1}}}      Log On


   



Question 245808: Use Descartes' Rule of Sign to determine how many positive and negative zeros each polynomial function may have.
f%28x%29+=+-6x%5E5+%2B+x%5E4+%2B+5x%5E3+%2B+x+%2B+1

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

First count the sign changes of f%28x%29=-6x%5E5%2Bx%5E4%2B5x%5E3%2Bx%2B1

From -6x%5E5 to x%5E4, there is a sign change from negative to positive

From x%5E4 to 5x%5E3, there is no change in sign

From 5x%5E3 to x, there is no change in sign

From x to 1, there is no change in sign

So there is 1 sign change for the expression f%28x%29=-6x%5E5%2Bx%5E4%2B5x%5E3%2Bx%2B1.

So there is 1 positive zero.



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f%28-x%29=-6%28-x%29%5E5%2B%28-x%29%5E4%2B5%28-x%29%5E3%2B%28-x%29%2B1 Now let's replace each x with -x


f%28-x%29=6x%5E5%2Bx%5E4-5x%5E3-x%2B1 Simplify


Now let's count the sign changes of f%28-x%29=6x%5E5%2Bx%5E4-5x%5E3-x%2B1

From 6x%5E5 to x%5E4, there is no change in sign

From x%5E4 to -5x%5E3, there is a sign change from positive to negative

From -5x%5E3 to -x, there is no change in sign

From -x to 1, there is a sign change from negative to positive

So there are 2 sign changes for the expression f%28-x%29=6x%5E5%2Bx%5E4-5x%5E3-x%2B1.

So there are 2 or 0 negative zeros