Problem 10 - College Algebra

Question

Solve each polynomial equation by factoring and then using the zero-product principle. $$3 x^{4}-81 x$$

Step-by-Step Solution

Answer

The solutions to the equation are: \(x = 0\), \(x = 9\), \(x = -9\) and \(x = \pm 3i\sqrt{3}\)

Step 1: Rearrange the equation

The given equation is \(3x^{4} - 81x = 0\). To factorize it, let's rearrange it as: \(x(3x^{3} - 81) = 0\)

Step 2: Factorize the equation

Now, factorize the equation to its simplest form: \(x[3(x - 9)(x + 9)(x^{2} + 9)] = 0\)

Step 3: Apply the zero product property

Applying the zero-product property, which states that if a product of factors is zero, then at least one of the factors must be zero. Therefore, setting each factor to equal zero gives, \(x = 0, x - 9 = 0, x + 9 = 0\) and \(x^{2} + 9 = 0\). From these equations, we can solve for \(x\).

Key Concepts

Factoring

Factoring is the process of breaking down a complex expression into simpler components called factors. In the context of polynomial equations, factoring helps us express an equation as a product of its factors, which makes it easier to solve. For example, in the given polynomial equation \(3x^4 - 81x = 0\), we begin by identifying the greatest common factor, which is \(3x\). By factoring out \(3x\), we simplify the equation to \(3x(x^3 - 27) = 0\).

• Identify common factors in terms of coefficients and variables.
• Factor out any common factors to simplify the expression.

In more complicated polynomial equations, additional techniques like grouping or special formulas may be needed to fully factor the equation. For our solution, further factoring gives the expression \(x[3(x - 9)(x + 9)(x^2 + 9)] = 0\), showing it as a product of factors, which is the key to the next step.

Zero-Product Property

The zero-product property is crucial when solving polynomial equations that involve factoring. It states that if the product of several factors is zero, then at least one of the factors must be zero. This property allows us to "break up" the equation into simpler parts and solve each one separately. For the polynomial equation \(x[3(x - 9)(x + 9)(x^2 + 9)] = 0\), we apply this property by setting each factor equal to zero:

• \(x = 0\)
• \(3 = 0\) - not possible, ignore.
• \(x - 9 = 0\)
• \(x + 9 = 0\)
• \(x^2 + 9 = 0\)

Thus, the zero-product property transforms the original equation into several simpler equations that we can solve individually.

Roots of an Equation

The roots of an equation are the solutions that satisfy the equation, making it true (i.e., resulting in zero when substituted back into the equation). For the polynomial equation \(3x^4 - 81x = 0\), after applying the zero-product property, we find several simple equations to solve.

• Solving \(x = 0\) gives the root \(x = 0\).
• Solving \(x - 9 = 0\) gives the root \(x = 9\).
• Solving \(x + 9 = 0\) gives the root \(x = -9\).

These values of \(x\) are called the roots because they are the values where the equation equals zero. Identifying roots is important in determining where a polynomial function intersects the x-axis on a graph.

Complex Solutions

Complex solutions occur when solving equations that lead to the square root of a negative number, typically when rational real-number solutions do not exist. In this exercise, we encounter the equation \(x^2 + 9 = 0\) as one of the factors. To solve for \(x\), rearrange this equation as \(x^2 = -9\). Since there are no real numbers whose square is negative, we introduce the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus,

• \(x = \pm \sqrt{-9} = \pm 3i\).

These complex solutions, \(x = 3i\) and \(x = -3i\), represent solutions not on the real number line but in the complex number plane. Identifying complex solutions is important as it shows the equation can be solved beyond real numbers, enhancing our understanding of polynomial behavior in complex domains.