Write a polynomial function of least degree with rational coefficients so that ​P(x)=0 has the given root.\

Question

2-7i
p(x)= __x^2-__x +__

Answer 1

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