# 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

Jan 2, 2018

The equation has $3$ real roots.

#### Explanation:

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

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 $a {x}^{3} + b {x}^{2} + c x + d = 0$, the discriminant will be:
$\Delta = 18 a b c d - 4 {b}^{3} d + {b}^{2} {c}^{2} - 4 a {c}^{3} - 27 {a}^{2} {d}^{2}$

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.