what's a monic polynomial with roots 2 and 4?

Answer 1

Start with:
x = 2
x = 4

These can be rewritten as:
x - 2 = 0
x - 4 = 0

Now you can multiply the two:
(x - 2)(x - 4) = 0

Expand using FOIL:
x² - 2x - 4x + 8 = 0

Combine the x terms:
x² - 6x + 8 = 0

(This is a monic polynomial because the coefficient on the x² is 1.)

Answer 2

What's the degree of the polynomial? If we don't know that, there are infinitely many such polynomials.

Monic means that the leading coefficient is 1. Roots are the zero points.

So a monic polynomial with roots 2 and 4 would be of the form (x-2)^a (x-4)^b where a and b are natural numbers.

For example:

(x-2) (x-4) = x² - 6x +8 (this is the only such polynomial of degree 2)

Answer 3

(x -2)(x - 4)

x^2 - 6x + 8