Problem 41 - Algebra 2

Question

Find the roots of \(x(x+6)(x-5)=0\)

Step-by-Step Solution

Answer

The roots are \(x = 0\), \(x = -6\), and \(x = 5\).

Step 1: Understand the Problem

We need to find the roots of the equation \(x(x+6)(x-5)=0\). These roots are the values of \(x\) that satisfy the equation.

Step 2: Apply the Zero Product Property

According to the zero product property, if the product of several factors is zero, then at least one of the factors must be zero. So, set each factor equal to zero: \(x = 0\), \(x+6 = 0\), and \(x-5 = 0\).

Step 3: Solve for Each Root

Solve the equations from Step 2:1. \(x = 0\)2. \(x + 6 = 0\) becomes \(x = -6\)3. \(x - 5 = 0\) becomes \(x = 5\)

Step 4: Collect the Roots

The solutions to the equation are the values found in Step 3. Therefore, \(x = 0\), \(x = -6\), and \(x = 5\).

Key Concepts

Roots of Polynomial

A polynomial equation represents a mathematical expression involving variables, coefficients, and exponents. The roots of a polynomial are the values of the variable that satisfy the equation, making it equal to zero. For instance, in the equation \(x(x+6)(x-5)=0\), the roots are the values of \(x\) at which the expression becomes zero.

To find the roots, you first rewrite the polynomial in a format that highlights its factors. Once the factors are revealed, the Zero Product Property—a key tool—comes into play. It states that if a product of factors equals zero, at least one of those factors must be zero. So, for our equation:

• \(x = 0\)
• \(x+6 = 0\) implies \(x = -6\)
• \(x-5 = 0\) implies \(x = 5\)

These roots \(x = 0, -6, 5\) are where the polynomial crosses the x-axis on a graph. Understanding this concept helps you solve similar equations efficiently.

Factoring Equations

Factoring is a fundamental technique used in algebra to simplify polynomials and make them easier to solve. It involves breaking down a polynomial into a product of its simpler, irreducible factors. In the equation \(x(x+6)(x-5)=0\), factoring has already been applied, since it is presented as a product of linear factors:

• \(x\)
• \(x+6\)
• \(x-5\)

Each factor represents a potential root of the equation. By setting each factor equal to zero, you can solve for the roots. This method is particularly useful in solving quadratic equations, where factoring translates a potentially complex problem into a set of simpler calculations. While some polynomials may need methods like completing the square or using the quadratic formula, the Zero Product Property offers a quick solution when the expression is already factored.

Solving Quadratic Equations

Solving quadratic equations is one of the critical skills in algebra, as these equations appear frequently in many scientific disciplines. A quadratic equation generally takes the form \(ax^2 + bx + c = 0\), and solving it means finding the values of \(x\) that satisfy this condition.

In an equation like \(x(x+6)(x-5)=0\), although not presented in the standard quadratic form, it essentially poses a similar challenge. The factored form makes the use of the Zero Product Property possible, streamlining the process of finding the roots without expanding the equation to its quadratic form. Otherwise, several techniques are available for solving quadratic equations, including:

• Factoring, which is straightforward if the equation is factorable.
• Completing the square, a method that rewrites the equation in a form that makes it easier to find the roots.
• The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), applicable to any quadratic equation.

When equations can be quickly factored like the given problem, applying the Zero Product Property allows students to bypass more complex algebraic manipulations. Understanding these methods provides a wide array of tools for tackling different forms of quadratic equations.