Question

One solution of the equation \(x^{3}+5 x^{2}+5 x-2=0\) is -2 . Find the sum of the remaining solutions.

Short answer

The sum of the remaining solutions is -3.

Step-by-step solution

- Understand Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For a cubic equation of the form, x^3 + ax^2 + bx + c = 0, the sum of the roots taken one at a time is given by -a (the opposite of the coefficient of the x^2 term).

- Identify Polynomial Coefficients

Given the polynomial equation x^3 + 5x^2 + 5x - 2 = 0, the coefficients are: a = 5, b = 5, c = -2.

- Calculate Sum of All Roots

Using Vieta's formulas, the sum of all the roots of x^3 + 5x^2 + 5x - 2 = 0 is found by taking the negative of the coefficient of the x^2 term. Therefore, the sum of all the roots is -5.

- Subtract Known Root

One known root is -2. To find the sum of the remaining roots, subtract the known root from the total sum of the roots. Sum of remaining roots = -5 - (-2) = -5 + 2 = -3.

Key concepts

Polynomial Roots

Polynomial roots are the values of the variable that make the polynomial equal to zero. For example, if we have a polynomial equation like \(x^3 + 5x^2 + 5x - 2 = 0\), the roots are the values of \(x\) that satisfy this equation. In simpler terms, roots are the solutions to the polynomial equation. Finding the roots involves factoring the polynomial or using various mathematical formulas and theorems, such as Vieta's formulas.

Sum of Roots

The sum of the roots of a polynomial can be determined without finding the individual roots. This is possible thanks to Vieta's formulas. According to Vieta's formulas, for a cubic polynomial of the form \(x^3 + ax^2 + bx + c = 0\), the sum of the roots is given by \(-a\), which is the negative of the coefficient of the \(x^2\) term. Let's take our example \(x^3 + 5x^2 + 5x - 2 = 0\). Here, the coefficient of \(x^2\) is 5, so the sum of the roots is \(-5\). Even if we don't know the individual roots, just knowing the coefficients allows us to compute their sum.

Cubic Equation

A cubic equation is a polynomial equation of degree three. It has the general form \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a \eq 0\). Cubic equations are more complex than quadratic equations and can have up to three real roots. Solving a cubic equation usually involves:

• Identifying at least one root through techniques like trial and error, or synthetic division.
• Using the found root to factorize the polynomial, reducing it to a quadratic equation.
• Solving the remaining quadratic equation to find the other roots.

Consider the example \(x^3 + 5x^2 + 5x - 2 = 0\). Knowing that one of the roots is -2 helps significantly. We can subtract this known root from the sum of all roots found via Vieta's formulas to find the sum of the remaining roots.