Three roots of the polynomial equation x^4 - 4x^3 - 2x^2 + 12x + 9 = 0 are 3, -1, and -1. Explain why the forth root must be a real number. Find the forth root.

Hello,

This polynomial function has maximum 4 roots as the degree is 4

and their coefficients are all reals so if one root is a complex number, its conjugate is a root too.

As we know already 3 roots and none of them is a complex number, the fourth one is a real number

And their product is 9 = 3 * 1 * (-1) * (-3) as

(The constant term is 9, the product of the roots)

so the last root is -3

Thanks

ye

step-by-step explanation:

δabc ≅ δdcb by sas because

db ≅ ac . . given (side)

bc ≅ bc . . reflexive property (side)

∠dbc ≅ ∠ acb . . alternate interior angles where transversal bc crosses parallel lines bd and ac (included angle)