Question

According to the rational root theorem, which is not a possible rational root of x3 + 8x2 x 6 = 0?

Answer 1

The rational root theorem says that if we have a leading coefficient of 1, any possible rational roots must be factors of the trailing term.

Notice that if we multiply two binomials:
\[(x+a)(x+b) = x(x+b)+a(x+b) = x^2 + bx + ax + ab = x^2+(a+b)x + ab\]the trailing term is just the product of the trailing terms of the binomials we multiplied. If we multiply that by another binomial:
\[(x+c)(x^2+(a+b)x + ab) = x^3 + (a+b)x^2 + abx + cx^2 + c(a+b)x + abc\]Again, the trailing term is just the product of the trailing terms. And because the polynomial can be written in factored form\[P(x) = (x-r_1)(x-r_2)...(x-r_n)=0\]where \(r_1,r_2,...r_n\) are the roots, any rational root must be a factor of that trailing term.