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Question 1039103: Show Work
Use the intermediate value theorem to show that the polynomial function has a zero in the given interval.
f(x)=x^5-x^4+4x^3-4x^2-20x+18
[1.5,1.8]
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Plug in x = 1.5 to get...
f(x)=x^5-x^4+4x^3-4x^2-20x+18
f(1.5)=(1.5)^5-(1.5)^4+4(1.5)^3-4(1.5)^2-20(1.5)+18
f(1.5)=7.59375-5.0625+4(3.375)-4(2.25)-20(1.5)+18
f(1.5)=7.59375-5.0625+13.5-9-30+18
f(1.5)= -4.96875
So when input is x = 1.5, the output is y = -4.96875
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Plug in x = 1.8 to get...
f(x)=x^5-x^4+4x^3-4x^2-20x+18
f(1.8)=(1.8)^5-(1.8)^4+4(1.8)^3-4(1.8)^2-20(1.8)+18
f(1.8)=18.89568-10.4976+4(5.832)-4(3.24)-20(1.8)+18
f(1.8)=18.89568-10.4976+23.328-12.96-36+18
f(1.8)= 0.76608
So when input is x = 1.8, the output is y = 0.76608
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To recap,
when we plug in x = 1.5, we get y = -4.96875
when we plug in x = 1.8, we get y = 0.76608
Take note how y changes from negative to positive as we move from x = 1.5 to x = 1.8. So y has to be zero somewhere between those two x values. This proves there is a root between x = 1.5 to x = 1.8
Side Notes:
- The function is assumed to be continuous.
- The term "root" is another term for "zero" or "x-intercept"
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