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# Let P(x) be a polynomial whose degree is 6. If P(n) =1/n for n = 1, 2, 3, 4, 5, 6, 7, compute the value P(8).

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Let P(x) be a polynomial whose degree is 6. If P(n) =1/n for n = 1, 2, 3, 4, 5, 6, 7, compute the value P(8).

Feb 15, 2021

#1
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This is DEFINITELY   tedious  to  solve!!!

The form is

P(x)   = ax^6  + bx^5  + cx^4  + dx^3  + ex^2  + fx  +  g

Using a computer algebra system, we  have   that

a  =1/ 5040

b  = -1/ 180

c  = 23/ 360

d   =  -7/18

e   = 967/720

f   =  -469/180

g =  363/140

P(x)    =  x^6/5040 - x^5/180 +23x^4/360 -7x^3/18+967x^2/720-469/180x +363/140

P(8)   = 1/4

Feb 15, 2021
#2
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`solve( a*1^6 + b*1^5 + c*1^4 + d*1^3 + e1*1^2 + f*1 + g=1/1, a*2^6 + b*2^5 + c*2^4 + d*2^3 + e1*2^2 + f*2 + g=1/2, a*3^6 + b*3^5 + c*3^4 + d*3^3 + e1*3^2 + f*3 + g=1/3, a*4^6 + b*4^5 + c*4^4 + d*4^3 + e1*4^2 + f*4 + g=1/4, a*5^6 + b*5^5 + c*5^4 + d*5^3 + e1*5^2 + f*5 + g=1/5, a*6^6 + b*6^5 + c*6^4 + d*6^3 + e1*6^2 + f*6 + g=1/6, a*7^6 + b*7^5 + c*7^4 + d*7^3 + e1*7^2 + f*7 + g=1/7 )`

`(1/5040)*8^6 + -((1/180))*8^5 + (23/360)*8^4 + -((7/18))*8^3 + (967/720)*8^2 + -((469/180))*8 + (363/140)`

May 6, 2021