If #(3-sqrt2)# and #(2+sqrt2)# are two of the roots of a fourth-degree polynomial with integer coefficients, what is the product of the other two roots?

1 Answer
Dec 25, 2016

Answer:

#4-sqrt(2)#

Explanation:

The two other roots will be the radical conjugates of the given ones, namely:

#(3+sqrt(2))# and #(2-sqrt(2))#

So (using FOIL) their product is:

#(3+sqrt(2))(2-sqrt(2)) = overbrace(3*2)^"First"+overbrace(3(-sqrt(2)))^"Outside"+overbrace((sqrt(2))2)^"Inside"+overbrace((sqrt(2))(-sqrt(2)))^"Last"#

#color(white)((3+sqrt(2))(2-sqrt(2))) = 6-3sqrt(2)+2sqrt(2)-2#

#color(white)((3+sqrt(2))(2-sqrt(2))) = 4-sqrt(2)#