+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).
+128475 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
