I don't know how to do this.

Question

Which describes the number and type of roots of the equation x^3 - 4x^2 + 50x + 7 = 0?

a. 1 positive, 2 negative
b. 2 positive, 1 negative
c. 3 negative
d. 3 positive.

Answer 1

Let X1, X2, X3 are roots of the equation,
then
X1+X2+X3=4
X1*X2+X1*X3+X2*X3=50
X1*X2*X3=-7
We can choose only b.

Answer 2

I don't know of any easy way to do this, so I used a website to provide a numerical solution.

None of the choic es are correct. Two of the roots are complex. The other is negative (-0.138414..)

Make sure you copied the problem correctly.

Answer 3

There are two changes in the sign of the coefficients, so we have either 2 positive roots, or none.

Also, by negating the odd-powered terms, there is one change of sign, so we have 1 negative root. This invalidates choices a, c and d.

Without actually solving the equation, the choice is either
(b), or
"none of the above", i.e. 2 complex and one negative root (as confirmed by drwls's solution).

Another way to see that there are 2 complex roots without solving the equation is to look at
f'(x)=0, or
3x^2-8x+50=0
which does not have real roots implying no maximum nor minimum. Therefore f(x) is monotonically increasing, and therefore has only one real root.