Do You Use the Formula -b/a to Find the Sum of the Roots in a Cubic Equation?

When dealing with polynomial equations, determining the sum of the roots is an essential step in understanding their properties. For a quadratic equation, the sum of the roots can be found using the formula -b/a. However, this formula is not directly applicable to cubic equations as it is for quadratic equations. In this article, we will explore how to find the sum of the roots of a cubic equation using both a direct method and Vieta's formulas.

Understanding the Sum of Roots in Quadratic Equations

For a quadratic equation in the standard form ax^2 bx c 0, the sum of the roots can be found using the formula -b/a. This formula is derived from the properties of polynomials and is a well-established result in algebra.

Sum of Roots in Cubic Equations

For cubic equations, the sum of the roots is instead given by -p, where p is the coefficient of the quadratic term in the equation x^3 px^2 qx r 0. This result is derived from Vieta's formulas, which provide a set of relationships between the coefficients of a polynomial and its roots.

Example: Finding the Sum of the Roots of the Equation (x^3 - 1 0)

The equation x^3 1 0 can be rewritten as x^3 ^2 - 1 0. Here, the coefficient p of the quadratic term (x^2) is 0. According to Vieta's formulas, the sum of the roots of this equation is -p -0 0.

First Method for Solving

In the first method, we start by noting that (x^3 -1), which has the solution (x -1). We can then write:

[x^3 - 1 0]

Since the polynomial has degree 3, it has 3 roots, which we can denote as (-1, r, s). Using the Fundamental Theorem of Algebra, we can factor the polynomial as:

[x^3 - 1 (x 1)(x - r)(x - s)]

Dividing (x^3 - 1) by (x 1), we get the quadratic equation (x^2 - x - 1 0). We know that:

[rs 1]

Thus, the sum of the roots (-1 r s) is:

[-1 rs -1 1 0]

Second Method Using Vieta's Rules

The second method involves using Vieta's rules for polynomials. For any polynomial of degree (n), the sum of the roots is given by (-frac{b}{a}), where (b) is the coefficient of the (x^{n-1}) term and (a) is the leading coefficient.

Inspecting the polynomial fx x^3 - 1, we see that the coefficient of the (x^2) term is 0. Therefore, the sum of the roots is:

[-frac{0}{1} 0]

General Form of a Cubic Equation

A general cubic equation is given by:

[ax^3 bx^2 cx d 0]

with roots (r, s, t). The equation factors as:

[ax^3 bx^2 cx d a(x - r)(x - s)(x - t) ax^3 - a(r s t)x^2 a(rs rt st)x - arst]

Equating respective coefficients, we get:

[b -ar(r s t), quad c a(rs rt st), quad d -arst]

From these relationships, we can derive:

[-frac{b}{a} r s t, quad frac{c}{a} rs rt st, quad -frac{d}{a} rst]

These are Vieta's formulas for a cubic equation. The formula -b/a is indeed used to find the sum of the roots.

Conclusion

In conclusion, both methods demonstrate that for a cubic equation, the sum of the roots can be found using Vieta's formulas, specifically (-p), where (p) is the coefficient of the quadratic term. For the specific equation (x^3 - ^2 - 1 0), the sum of the roots is indeed 0.

Understanding these formulas and methods can greatly assist in solving and analyzing cubic equations, making it a valuable tool in algebra and polynomial theory.