SOLUTION: Use the Intermediate Value Theorem to determine if {{{P(x)=2x^5-7x+1}}} has a zero in the interval [1,2]

Algebra ->  Rational-functions -> SOLUTION: Use the Intermediate Value Theorem to determine if {{{P(x)=2x^5-7x+1}}} has a zero in the interval [1,2]      Log On


   

Question 149765: Use the Intermediate Value Theorem to determine if P%28x%29=2x%5E5-7x%2B1 has a zero in the interval [1,2]
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To use the intermediate value theorem, simply evaluate the endpoints of the interval.


Let's evaluate the left endpoint x=1


P%28x%29=2x%5E5-7x%2B1 Start with the given equation.


P%281%29=2%281%29%5E5-7%281%29%2B1 Plug in x=1.


P%281%29=2%281%29-7%281%29%2B1 Raise 1 to the 5th power to get 1.


P%281%29=2-7%281%29%2B1 Multiply 2 and 1 to get 2.


P%281%29=2-7%2B1 Multiply -7 and 1 to get -7.


P%281%29=-4 Combine like terms.


So the function value at x=1 is P%281%29=-4. This makes the point (1,-4) which tells us that the y-coordinate is negative.

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Now let's evaluate the right endpoint x=2


P%28x%29=2x%5E5-7x%2B1 Start with the given equation.


P%282%29=2%282%29%5E5-7%282%29%2B1 Plug in x=2.


P%282%29=2%2832%29-7%282%29%2B1 Raise 2 to the 5th power to get 32.


P%282%29=64-7%282%29%2B1 Multiply 2 and 32 to get 64.


P%282%29=64-14%2B1 Multiply -7 and 2 to get -14.


P%282%29=51 Combine like terms.



So the function value at x=2 is P%282%29=51. This makes the point (2,51) which tells us that the y-coordinate is positive.



Since the sign of the y-coordinate transitioned from negative to positive on the interval [1,2], this means that the graph must have crossed the x-axis somewhere in the interval. So there is a zero in the interval [1,2].