SOLUTION: 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]

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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