Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Find the real solution(s) of the polynomial equation. Check your solutions. \(5 x^{3}+30 x^{2}+45 x=0\)

Step-by-Step Solution

Answer

The real solutions to the equation \(5 x^{3}+30 x^{2}+45 x=0\) are \(x=0\) and \(x=-3\).

Step 1: Factor out the Greatest Common Divisor (GCD)

The GCD in the equation \(5 x^{3}+30 x^{2}+45 x=0\) is \(x\). Factor this out to get \(x(5x^2 + 30x + 45) = 0\).

Step 2: Simplify the equation

Simplify the equation to remove any common factors within the parenthesis. We can factor out 5 from the terms inside the parenthesis to get: \(x*5(x^2+6x+9)=0\) or \(5x(x+3)^2 = 0\).

Step 3: Apply Zero-Product Property

The zero-product property states that if the product of multiple factors is zero, then at least one of the factors must be zero. Set each factor to zero and solve for \(x\). These solutions will be the real solutions to the equation. So, we solve \(5x = 0\) and \((x+3)^2 = 0\). The solutions are \(x=0\) and \(x=-3\).

Key Concepts

Factoring

Factoring is an essential technique in solving polynomial equations. It involves breaking down a complex expression into simpler parts called "factors." In the given polynomial equation, \(5x^3 + 30x^2 + 45x = 0\), we start by factoring out the common elements in all terms. This process simplifies the equation, making it easier to solve.

The first step is identifying the Greatest Common Divisor (GCD) of the terms \(5x^3\), \(30x^2\), and \(45x\). Here, each term contains the variable \(x\), so \(x\) is a common factor for all the terms. By factoring \(x\) out, the equation simplifies to \(x(5x^2 + 30x + 45) = 0\).

After isolating the \(x\), look at the remaining expression inside the parentheses. The expression \(5x^2 + 30x + 45\) itself can be factored further. In our example, we identify that \(5\) is a common factor of all coefficients, leading to \(x \cdot 5(x^2 + 6x + 9) = 0\). Therefore, further simplification yields \(5x(x+3)^2 = 0\), effectively breaking the polynomial into its simplest components.

• Identify and factor out the greatest common factor.
• Factor the remaining polynomial.
• Continue simplifying until fully reduced.

Zero-Product Property

The Zero-Product Property is a simple yet powerful tool in solving polynomial equations. This principle states that if a product of multiple factors equals zero, then at least one factor must be zero.

In our problem, after factoring, we have the simplified polynomial equation \(5x(x+3)^2 = 0\). According to the Zero-Product Property, either \(5x = 0\), \((x+3) = 0\), or \((x+3)\) has been squared, thus contributing to zero once squared.

By applying this property, we set each factor containing \(x\) to zero and solve for \(x\):

• Set \(5x = 0\), leading to the solution \(x = 0\).
• Set \((x+3)^2 = 0\). Solving \(x+3 = 0\), we find \(x = -3\).

Thus, our solutions are \(x = 0\) and \(x = -3\). This highlights how setting each factor to zero allows us to find every possible solution of the equation.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is an important concept when simplifying polynomial equations. It refers to the highest factor shared by two or more numbers or terms.

In polynomial expressions, finding the GCD enables us to streamline the equation by reducing it to simpler terms.

In the equation \(5x^3 + 30x^2 + 45x = 0\), each term shares a common variable factor \(x\) and number factor, \(5\). Hence, \(5x\) is the Greatest Common Divisor for all terms.

• Identify shared numerical and variable factors.
• Factor out the GCD to simplify the polynomial.

By factoring out \(5x\), our equation takes on a simpler form \(5x(x^2 + 6x + 9) = 0\), making it easier to solve by further techniques like factoring completely or using the Zero-Product Property. Identifying and factoring out the GCD effectively reduces the complexity of solving polynomial equations.