Problem 13 - Algebra 2

Question

Solve each equation. State the number and type of roots. \(x^{3}+9 x=0\)

Step-by-Step Solution

Answer

The equation has one real root (0) and two complex roots (3i, -3i).

Step 1: Recognize the polynomial equation

The given equation is a cubic polynomial, which is written as \(x^3 + 9x = 0\). This is a homogeneous polynomial equation of degree 3.

Step 2: Factor the equation

To solve the equation, start by factoring out the common factor, which is \(x\). This gives us \(x(x^2 + 9) = 0\).

Step 3: Set each factor equal to zero

Using the zero-product property, set each factor equal to zero: \(x = 0\) and \(x^2 + 9 = 0\).

Step 4: Solve for real roots

From \(x = 0\), we immediately have a real root: \(x = 0\).

Step 5: Solve for complex roots

Now, solve \(x^2 + 9 = 0\). Subtract 9 from both sides to get \(x^2 = -9\). Taking the square root of both sides gives \(x = \pm i\sqrt{9}\) or \(x = \pm 3i\).

Step 6: State the type and number of roots

The equation \(x^3 + 9x = 0\) has one real root (\(x = 0\)) and two complex roots (\(x = 3i\) and \(x = -3i\)).

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions that include variables raised to whole number powers with constant coefficients. These equations are central to algebra and provide the foundation for many advanced mathematical concepts. In a general form, a polynomial equation can be expressed as \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\), where each \(a_i\) is a constant and \(n\) denotes the degree of the polynomial.

There are various types of polynomial equations, with degrees affecting their complexity:

• Linear equations have a single variable term and are of degree 1 (e.g., \(ax + b = 0\)).
• Quadratic equations are of degree 2 (e.g., \(ax^2 + bx + c = 0\)).
• Cubic equations, like in our exercise, have a maximum degree of 3 (e.g., \(x^3 + 9x = 0\)).

To solve a polynomial equation, you often need to factor them, apply various algebraic techniques, or use numerical methods for higher degrees. Understanding polynomial structure aids in recognizing root types and predicting behavior, critical skills in advanced math topics.

Zero-Product Property

The zero-product property is a fundamental rule in algebra that states if a product of multiple factors is zero, then at least one of the factors must be zero. This property is especially useful when solving polynomial equations that are already factored or can be factored easily.

For example, if we have a factored polynomial equation like \(x(x^2 + 9) = 0\), the zero-product property allows us to set each factor separately to zero:

• \(x = 0\)
• \(x^2 + 9 = 0\)

Solving each factor gives potential solutions to the original equation.
Using this property simplifies finding roots significantly, as it breaks down complex expressions into simpler terms to evaluate independently. It's important to practice factoring and apply this property confidently, as it is key in algebraic problem-solving and finding solutions to polynomial equations.

Complex Roots

Not all polynomial equations have real roots. Especially when dealing with higher-than-first-degree polynomials, complex roots often appear. A complex root contains an imaginary unit \(i\), which is defined by the property \(i^2 = -1\). This property leads to complex solutions where no real number satisfies the equation.

To illustrate, from our factored equation \(x^2 + 9 = 0\), we solve for complex roots by rearranging and using the imaginary unit:

• First, \(x^2 = -9\)
• Taking square roots, \(x = \pm \sqrt{-9}\)
• Thus, \(x = \pm 3i\)

These are the complex roots, \(3i\) and \(-3i\). Understanding complex roots is essential, as they demonstrate the breadth of solution types for polynomials, provide insight into the behavior of functions, and ensure all solutions (real or complex) are considered. Often complex solutions occur in conjugate pairs when coefficients are real, reflecting their symmetry and preserving the real freedom of polynomial expressions.