Problem 11 - Algebra Form and Function

Question

Give all the solutions of the equations. $$ (x-1) x(x+3)=0 $$

Step-by-Step Solution

Answer

Answer: The solutions are x = 1, x = 0, and x = -3.

Step 1: Identify the factors of the equation

The given equation is: $$ (x-1) x(x+3)=0 $$ Notice that we have three factors: \((x-1)\), \(x\), and \((x+3)\).

Step 2: Find the zeros of each factor

To find all the solutions of the equation, we need to find the zeros of each factor. We set each factor equal to zero and solve for \(x\): Factor 1: \((x-1)=0\) $$ x-1=0 \Rightarrow x=1 $$ Factor 2: \(x=0\) $$ x=0 $$ Factor 3: \((x+3)=0\) $$ x+3=0 \Rightarrow x=-3 $$

Step 3: Write down the solutions

Now that we have found the zeros of all the factors, we can write down the solutions of the given equation: $$ x = 1,\ x = 0,\ x = -3 $$ These are all the solutions of the given equation.

Key Concepts

Factoring Polynomials

Factoring polynomials is a method used to express a polynomial as a product of its simpler polynomial factors. The primary reason to factor polynomials is to simplify solving equations or performing other algebraic operations involving them. When factoring polynomials, you're essentially breaking down a complex expression into smaller, more manageable parts.
Consider the polynomial equation given:

• \((x-1)x(x+3)=0\)

This equation is already factored, with its expression decomposed into three simpler factors: \((x-1)\), \(x\), and \((x+3)\). Each of these factors represents a potential root of the polynomial.
To factor more complex polynomials, techniques such as finding the greatest common factor, grouping, or using special formulas like the difference of squares may be necessary. In the given exercise, though, the equation has been provided in its factored form, which simplifies the process of finding solutions.

Solving Equations

Solving equations, particularly polynomial equations, involves finding the values of the variable that make the equation true. For the equation

• \((x-1)x(x+3)=0\)

this means finding the values of \(x\) that satisfy the equation. These are the solutions or roots of the equation.
In this scenario, solving the equation is simplified by the fact that it is already in its factored form. This allows us to directly apply the Zero Product Property to each factor to find the solution. The fundamental principle when solving is identifying the values of the variable that reduce each factor to zero, as achieving a product of zero mandates that at least one of the multiplicands is zero. It's a straightforward yet powerful tool in algebra, ensuring that even complex equations can be addressed methodically.

Zero Product Property

The Zero Product Property is a crucial concept in algebra that states if the product of two or more factors is zero, then at least one of the factors must be zero. This property is central to solving polynomial equations once they are in factored form.
Applying this property to our equation:

• \((x-1)x(x+3)=0\)

means setting each factor equal to zero:

• \(x-1=0\)
• \(x=0\)
• \(x+3=0\)

Solving these equations gives the solutions \(x=1\), \(x=0\), and \(x=-3\), respectively. This approach efficiently pinpoints the values of \(x\) that satisfy the original polynomial equation.
Understanding the Zero Product Property allows students to solve equations swiftly and with confidence, knowing exactly which steps to take to reach the correct answers.