Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, solve the following polynomial equations by grouping and factoring. $$ 2 x^{5}-14 x^{3}=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0, x = \sqrt{7}\), and \(x = -\sqrt{7}\).

Step 1: Find the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) in the polynomial. In the equation \(2x^5 - 14x^3 = 0\), both terms contain a common factor of \(2x^3\). Thus, the GCF is \(2x^3\).

Step 2: Factor Out the GCF

Factor \(2x^3\) out of the polynomial. After factoring, the equation becomes:\[2x^3(x^2 - 7) = 0\]

Step 3: Apply the Zero Product Property

According to the zero product property, if a product of factors equals zero, at least one of the factors must be zero. So, set each factor equal to zero and solve:1. \(2x^3 = 0\) 2. \(x^2 - 7 = 0\)

Step 4: Solve Each Factor for Zero

Solve the equations from the previous step:1. \(2x^3 = 0\): Divide both sides by 2, \(x^3 = 0\) Take the cube root of both sides, \(x = 0\) 2. \(x^2 - 7 = 0\): Add 7 to both sides, \(x^2 = 7\) Take the square root of both sides, \(x = \pm \sqrt{7}\)

Key Concepts

Greatest Common Factor

When working with polynomial equations like \(2x^5 - 14x^3 = 0\), the first step is often searching for the Greatest Common Factor (GCF). The GCF is the largest factor that divides all terms of the polynomial without leaving any remainder. This helps in simplifying the equation, making it easier to solve later.

To find the GCF in our equation, we look at each term: \(2x^5\) and \(-14x^3\). Both terms have a coefficient that is divisible by 2 and contain \(x\) as a common variable. The smallest power of \(x\) common to both is \(x^3\), so the GCF is \(2x^3\).

Extracting the GCF is like pulling out a common element that simplifies the expression, which is crucial for factoring in the next steps.

Factoring

Once we have identified the GCF, the next step is factoring out this common factor from the polynomial. This step is vital for simplifying the equation and reaching a form where we can easily identify solutions.

In our example, we factor out \(2x^3\) from the polynomial \(2x^5 - 14x^3\). It's like redistributing the terms for simplicity, resulting in:

• \(2x^5 - 14x^3 = 2x^3(x^2 - 7)\).

Factoring is a method used to express the equation as a product of simpler expressions, which is useful in solving the equations using other methods like the Zero Product Property.

Zero Product Property

The Zero Product Property is an essential principle when solving polynomial equations that have been factored. This property states that if the product of two or more factors is zero, at least one of the factors must be zero.

In our example, the factored equation is \(2x^3(x^2 - 7) = 0\). According to the Zero Product Property, we can set each factor equal to zero:

• \(2x^3 = 0\)
• \(x^2 - 7 = 0\)

This approach turns a possibly complex equation into smaller, simpler ones that are easier to solve. It’s a powerful tool that helps find all possible values that make the original equation true.

Solving Polynomial Equations

Using the Zero Product Property, we have broken down our polynomial equation into simpler problems. Solving these individual equations is the final step in finding all possible solutions to the original polynomial equation.

We solve each factor from the step above:

• For \(2x^3 = 0\), divide by 2 to simplify to \(x^3 = 0\), then take the cube root to find \(x = 0\).
• For \(x^2 - 7 = 0\), add 7 to both sides to get \(x^2 = 7\), then take the square root, giving \(x = \pm \sqrt{7}\).

So, the solutions to the polynomial equation \(2x^5 - 14x^3 = 0\) are \(x = 0\) and \(x = \pm \sqrt{7}\).

This systematic approach of breaking down, factoring, and solving highlights the value of each concept in the context of solving polynomial equations.