Write a polynomial function of least degree with rational coefficients so that P(x) = 0 has the given root 2-7i

Question

Answer

P(x)=x^2-(answer here)x+(answer here)

Answer 1

If a polynomial has the root 2 - 7i, then it also has the root 2 + 7i by the complex conjugate root theorem.

Multiply the factors together [x - (2 + 7i)][x - (2 - 7i)] ----> (x - 2 - 7i)(x - 2 + 7i)

= x² - 2x + 7ix - 2x + 4 - 14i - 7ix + 14i - 49i²

Simplify:

= x² -11x - 49(-1)

= x² -11x + 49

Answer 2

Any time you need rational coefficients and have an imaginary or complex root, you'll need to multiply by a factor that eliminates the imaginary number, i.

In this case, (a - bi) gets multiplied by (a + bi).

Use FOIL (First, Outside, Inside, Last) or the Box Method to distribute. Your Outside and Inside terms should cancel out.

(a - bi)(a + bi) = a² +abi - abi -b²i²

Then i² = -1, so it becomes

a² - b²(-1)

= a² + b²

And if there aren't any other variables, you can simplify one step further.