# Given the roots of the equation x = 1, 4, 2, what is the simplest equation.

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If the roots of an equation in x are 1, 2, 4:

(x - 1)(x - 2)(x - 4) = 0

=> (x^2 - x - 2x + 2)(x - 4) = 0

=> (x^2 - 3x + 2)(x - 4) = 0

=> x^3 - 4x^2 - 3x^2 + 12x + 2x - 8 = 0

=> x^3 - 7x^2 + 14x - 8 = 0

**The required equation is x^3 - 7x^2 + 14x - 8 = 0**

(x - 1)(x - 2)(x - 4) = 0

(x^2 - x - 2x + 2)(x - 4) = 0

combine:

(x^2 - 3x + 2)(x - 4) = 0

foil:

x^3 - 4x^2 - 3x^2 + 12x + 2x - 8 = 0

combine like terms:

x^3 - 7x^2 + 14x - 8 = 0

If the roots are x = 1, 4, 2,

it means that you work your problem backwards. Instead of factoring you start by the factored version.

x = 1 means (X-1)

x = 4 means (x - 4 )

x = 2 means (x - 2)

(x - 1)(x - 2)(x - 4) = 0

now you foil the first 2 parentheses:

(x - 1)(x - 2) l (x - 4) = 0

(x^2 - x - 2x + 2)(x - 4) = 0

now foil again with the other parentheses:

(x^2 - 3x + 2)(x - 4) = 0

x^3 - 4x^2 - 3x^2 + 12x + 2x - 8 = 0

combine like terms:

x^3 - 7x^2 + 14x - 8 = 0