How many roots do the following equations have? -12x^2 -25x +5 +x^3 =0

Select one:
a. 4
b. 2
c. 3
d. 5

1 Answer
Jan 2, 2018

Answer:

The equation has #3# real roots.

Explanation:

First we write the polynomial in standard form:
#x^3-12x^2-25x+5#

Since the polynomial is of degree three, the fundamental theorem of algebra tells us that it has #3# complex roots. Since (non-real) complex roots to functions with real coefficients only come in conjugate pairs, we can deduce that at least one of these #3# complex roots is real.

I presume the question is asking for the number of real roots, so to determine whether the function has #1# or #3# real roots, I will look at the discriminant of the cubic equation.

If you have a cubic equation in the form #ax^3+bx^2+cx+d=0#, the discriminant will be:
#Delta=18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2#

If #Delta>0#, we get three distinct roots, if #Delta=0#, we get real roots but one of them will be a multiple root, and finally if #Delta<0#, we will have #1# real root and #2# complex roots.

In our case, we get #Delta=213385#, so we can conclude that we have #3# real roots.