+318 Use the intermediate value theorem to show that the polynomial P(x) has a real zero in the interval [2,3]. Approximate this zero to two decimal places
P(x)=2x^4-4x^3-9
+118703
Use the intermediate value theorem to show that the polynomial P(x) has a real zero in the interval [2,3]. Approximate this zero to two decimal places
P(x)=2x^4-4x^3-9
I do not know offhand what intermediate value theorem is but I expect I can answer this question.
P(2)=2∗23−4∗23−9P(2)=16−32−9P(2)=−25 P(3)=2∗34−4∗33−9P(3)=162−108−9p(3)>0
Since P(2) is negative and P(3) is positive there must be a root between x=2 and x=3
Try P(2.5)
P(x)=2x4−4x3−9P(2.5)=2∗2.54−4∗2.53−9P(2.5)=6.625P(2.5)>0
So it must be between 2 and 2.5
The midpoint of 2 and 2.5 is 2.25
P(x)=2x4−4x3−9P(2.25)=2∗2.254−4∗2.253−9P(2.25)=−3.3P(2.25)<0
So the root is between 2.25 and 2.50 The midpoint of these is 2.375
P(x)=2x4−4x3−9P(2.375)=2∗2.3754−4∗2.3753−9P(2.375)=1.047P(2.375)>0
So the root is between 2.25 and 2.375. The average of these is 2.3125
P(x)=2x4−4x3−9P(2.3125)=2∗2.31254−4∗2.31253−9P(2.3125)=−1.27P(2.3125)<0
So the root is between 2.3125 and 2.375 The average of these is 2.34375 which is approx 2.344
P(x)=2x4−4x3−9P(2.344)=2∗2.3444−4∗2.3443−9P(2.344)=−0.139P(2.344)<0
So the root is between 2.344 and 2.375. The average of these is 2.3595 which is approx 2.360
P(x)=2x4−4x3−9P(2.360)=2∗2.3604−4∗2.3603−9P(2.360)=0.4638P(2.360)>0
So the root is between 2.360 and 2.344. The average of these is 2.352
P(x)=2x4−4x3−9P(2.352)=2∗2.3524−4∗2.3523−9P(2.352)=0.159P(2.352)>0
So the root is between 2.352 and 2.344. The average of these is 2.348
P(x)=2x4−4x3−9P(2.348)=2∗2.3484−4∗2.3483−9P(2.348)=0.00955P(2.348)>0
So the root is between 2.348 and 2.344. The average of these is 2.346
P(x)=2x4−4x3−9P(2.346)=2∗2.3464−4∗2.3463−9P(2.346)=−0.65P(2.346)<0
So the root is between 2.346 and 2.348.
So to 2 decimal places the root is approx 2.35
What an effort. Maybe you were meant to use a quicker method.
Coding:
P(2)=2*2^3-4*2^3-9\\
P(2)=16-32-9\\
P(2)=-25\\~\\
P(3)=2*3^4-4*3^3-9\\
P(3)=162-108-9\\
p(3)>0
P(x)=2x^4-4x^3-9\\
P(2.5)=2*2.5^4-4*2.5^3-9\\
P(2.5)=6.625\\
P(2.5)>0\\
P(x)=2x^4-4x^3-9\\
P(2.25)=2*2.25^4-4*2.25^3-9\\
P(2.25)=-3.3
\\P(2.25)<0
P(x)=2x^4-4x^3-9\\
P(2.375)=2*2.375^4-4*2.375^3-9\\
P(2.375)=1.047
\\P(2.375)>0
P(x)=2x^4-4x^3-9\\
P(2.3125)=2*2.3125^4-4*2.3125^3-9\\
P(2.3125)=-1.27
\\P(2.3125)<0
P(x)=2x^4-4x^3-9\\
P(2.344)=2*2.344^4-4*2.344^3-9\\
P(2.344)=-0.139
\\P(2.344)<0
P(x)=2x^4-4x^3-9\\
P(2.360)=2*2.360^4-4*2.360^3-9\\
P(2.360)=0.4638
\\P(2.360)>0
P(x)=2x^4-4x^3-9\\
P(2.352)=2*2.352^4-4*2.352^3-9\\
P(2.352)=0.159
\\P(2.352)>0
P(x)=2x^4-4x^3-9\\
P(2.348)=2*2.348^4-4*2.348^3-9\\
P(2.348)=0.00955
\\P(2.348)>0
