# Real Roots of Cubic Equation: $x^3+a^3x^2+b^3x+c^3=0$

• MHB
• anemone
In summary, a cubic equation is a type of polynomial equation with the highest degree of 3. It can have either 3 real roots or 1 real root, and it is not possible for it to have 2 real roots. To find the real roots, you can use Cardano's method or Ferrari's solution, which both involve solving a quadratic equation. The coefficients of a cubic equation have a direct relationship with its roots, such as the sum of the roots being equal to the negative coefficient of the quadratic term and the product of the roots being equal to the constant term.
anemone
Gold Member
MHB
POTW Director
An equation $x^3+ax^2+bx+c=0$ has three (but not necessarily distinct) real roots $t,\,u,\,v$. For what values of $a,\,b,\,c$ are the numbers $t^3,\,u^3,\,v^3$ roots of an equation $x^3+a^3x^2+b^3x+c^3=0$?

Let $P(x)=x^3+ax^2+bx+c$ with roots $t,\,u,\,v$ and $Q(x)=x^3+a^3x^2+b^3x+c^3$ whose roots are $t^3,\,u^3,\,v^3$ respectively. By the Viete formula, we have

$t+u+v=-a,\\tu+uv+vt=b,\\tuv=-c$ and

$t^3+u^3+v^3=-a^3,\\(tu)^3+(uv)^3+(vt)^3=b^3,\\(tuv)^3=-c^3$

Note that

$(t+u+v)^3=t^3+u^3+v^3+3(t+u+v)(tu+uv+vt)-3tuv$

which gives $-a^3=-a^3-3ab+3c$, or equivalently, $c=ab$. In this case $Q(x)$ has the form

$Q(x)=x^3+a^3x^2+b^3x+(ab)^3=(x+a^3)(x^2+b^3)$

This polynomial has a root $x=-a$ and for the other two roots we should have $b\le 0$. Thus the conditions are

$ab=c,\\ b\le 0$

## 1. What is a cubic equation?

A cubic equation is a polynomial equation of degree three, meaning that the highest power of the variable is three. It takes the form of ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and x is the variable.

## 2. How do you find the real roots of a cubic equation?

To find the real roots of a cubic equation, you can use the rational root theorem to determine potential rational roots. Then, you can use synthetic division or the quadratic formula to find the remaining roots. It is also helpful to graph the equation to visually see the roots.

## 3. What is the significance of the real roots of a cubic equation?

The real roots of a cubic equation represent the values of the variable that make the equation equal to zero. These roots can be used to solve real-world problems, such as finding the dimensions of a box with a given volume or determining the maximum profit of a business.

## 4. Can a cubic equation have more than three real roots?

No, a cubic equation can only have a maximum of three real roots. This is because a cubic equation is a polynomial of degree three, meaning it can only have three solutions at most.

## 5. How do you know if a cubic equation has no real roots?

If all three roots of a cubic equation are complex numbers, then the equation has no real roots. This means that the graph of the equation does not intersect the x-axis. Additionally, if the discriminant (b^2 - 4ac) is negative, then the equation has no real roots.

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