How do you find a polynomial function that has zeros 1+sqrt3, 1-sqrt3?
2 Answers
Explanation:
Since these are the zeros, we can make the following equation:
(x-(1+sqrt3))(x-(1-sqrt3)) = 0
Or
(x-1-sqrt3)(x-1+sqrt3) = 0
When we expand this, we get
x^2 - x + xsqrt3 - x + 1 - sqrt3 - xsqrt3 + sqrt3 - 3 = 0
Combining like terms:
color(blue)(ulbar(|stackrel(" ")(" "f(x) = x^2 - 2x - 2" ")|)
Explanation:
The simplest polynomial with distinct zeros
(x-alpha)(x-beta) = x^2-(alpha+beta)x+alphabeta
With
{ (alpha+beta = (1+sqrt(3))+(1-sqrt(3)) = color(red)(2)), (alphabeta = (1+sqrt(3))(1-sqrt(3)) = 1^2-(sqrt(3))^2 = 1-3 = color(blue)(-2)) :}
So a suitable polynomial function would be:
f(x) = x^2-color(red)(2)xcolor(blue)(-2)
Any polynomial function in