Question

Write a polynomial of least degree with rational coefficients and with the root

–15+10
`√(6\\)`

Answer 1

p(x) = x² +30x -375

Explanation:

When a quadratic has real rational coefficients, any irrational or complex roots come in conjugate pairs.

Factored form

A root of p means (x -p) is a factor of the polynomial. Here, we have roots of -15+10√6 and -15-10√6, so the factored form can be written ...

p(x) = (x -(-15 +10√6))(x -(-15 -10√6))

Using the factoring of the difference of squares, we can write this as ...

p(x) = (x +15)² -(10√6)²

Standard form

Expanding the factored form, we can write the polynomial as ...

p(x) = x² +30x +225 -600

p(x) = x² +30x -375