+0  
 
0
100
2
avatar+52 

Suppose P(x) is a polynomial of smallest possible degree such that:

- P(x) has rational coefficients

-\(P(-3) = P(\sqrt 7) = P(1-\sqrt 6) = 0\)

-P(-1) = 8

Determine the value of P(0).

 Jun 2, 2021
 #1
avatar+121056 
+2

Since  sqrt (7)   and  1 - sqrt (6)  are roots  then  so  are  -sqrt (7) and  1 + sqrt (6)

So....we  have a 5th degree polynomial

 

This polynomial  is

 

a ( x + 3) ( x- sqrt 7) ( x + sqrt 7) ( x - 1  + sqrt 6)  ( x - 1 - sqrt (6)

 

Simplifying this  we  gat

 

a (x^5 - x^4 - 18x^3  - 22x^2  + 77x  + 105)

 

And  since  we  know   that   P(-1)  =  8, we  can solve  this  for  "a"

 

8  =  a ( -1 + 3) (- 1 - sqrt 7)  (-1 + sqrt 7) ( -1 - 1 + sqrt (6))  ( -1 - 1 - sqrt (6))   simplify

 

8   = a  (24)

 

a =  (8/24)  =   1/3

 

The polynomial  is    (1/3)  ( x^5 + x^4 - 18 x^3 - 22 x^2 + 77 x + 105)

 

So.....P(0)  =     105 / 3    =   35

 

 

 

cool cool cool

 Jun 2, 2021
 #2
avatar+52 
0

This really helped, thanks!

ThanksForAllHelp  Jun 3, 2021

35 Online Users

avatar
avatar