Write a polynomial function of least degree with rational coefficients so that P(x)=0 has the given root.\
2-7i
p(x)= __x^2-__x +__
The given root is 2-7i. Since polynomial functions with rational coefficients always have complex conjugate roots, the conjugate of 2-7i is 2+7i.
By the conjugate root theorem, if a polynomial P(x) has rational coefficients and 2-7i is a root, then 2+7i must also be a root.
So, the polynomial function P(x) can be written as:
P(x) = (x - (2-7i))(x - (2+7i))
Expanding this expression, we get:
P(x) = (x - 2 + 7i)(x - 2 - 7i)
By using the difference of squares, we can simplify this to:
P(x) = (x - 2)^2 - (7i)^2
Simplifying further, we have:
P(x) = (x - 2)^2 - 49i^2
Since i^2 equals -1, we can replace i^2 with -1:
P(x) = (x - 2)^2 - 49(-1)
P(x) = (x - 2)^2 + 49
So, the polynomial function with the given root 2-7i is:
P(x) = (x - 2)^2 + 49
Therefore, p(x) = x^2 - 4x + 4 + 49
p(x) = x^2 - 4x + 53