Problem 24 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ n(n+1)(n-6)=0 $$

Step-by-Step Solution

Answer

The solutions are \(n = 0, -1,\) and \(6\).

Step 1: Understand the Zero Product Property

The problem is a polynomial equation in the form of a product of terms: \(n(n+1)(n-6)=0\). According to the Zero Product Property, the equation holds true if at least one of the terms in the product is equal to zero. So, to find the solutions, set each factor equal to zero separately.

Step 2: Solve for First Factor

Set the first factor equal to zero: \(n = 0\). This gives us one solution: \(n = 0\).

Step 3: Solve for Second Factor

Set the second factor equal to zero: \(n + 1 = 0\). Solve for \(n\) by subtracting 1 from both sides to obtain \(n = -1\). This gives us another solution: \(n = -1\).

Step 4: Solve for Third Factor

Set the third factor equal to zero: \(n - 6 = 0\). Solve for \(n\) by adding 6 to both sides to find \(n = 6\). This gives us the final solution: \(n = 6\).

Key Concepts

Polynomial Equations

Polynomial equations are expressions where multiple terms are summed together, each involving variables raised to a certain power. In this exercise, we are given a product of terms: the equation \( n(n+1)(n-6)=0 \). The expression defines a polynomial equation because it includes terms like \( n^2 \) and \( n^3 \). Recognizing polynomials is crucial, as it helps us decide the method to solve them.

• Definition: A polynomial is an expression with one or more terms, involving powers of variables and coefficients.
• Example of Polynomials: \( 3n^2 + 2n - 5 \) is another polynomial, where each term is separated by plus or minus.
• Understanding Polynomials: The degree of a polynomial is determined by the highest power of the variable present. In the given example, this polynomial is of the third degree due to the term \( n^3 \).

Understanding the structure of polynomials gives you the tools needed to select the correct solving technique.

Factoring

Factoring is a method used to solve polynomial equations by identifying and breaking down the equation into simpler terms called factors. In our given equation \( n(n+1)(n-6)=0 \), factoring is already done for us as it's already presented in a product form. But normally, you may need to factor a complex polynomial to get it into a solvable form.

• Purpose of Factoring: It simplifies equations, making them easier to handle, especially when setting each factor to zero.
• How to Factor: Look for common factors in terms, apply techniques like grouping, or use special products such as difference of squares.
• Example: Factoring \( n^2 - 6n \) could result in \( n(n-6) \), simplifying the process of setting each factor equal to zero.

Factoring forms the backbone for solving polynomial equations by converting complex equations into more manageable pieces.

Solving Equations

Solving equations involves finding the values of the variables that satisfy the equation. For a polynomial like \( n(n+1)(n-6)=0 \), this typically involves using the Zero Product Property. This property states that if a product of factors equals zero, at least one of the factors must be zero.

• Zero Product Property: It tells us that if \( ab = 0 \), then either \( a = 0 \), \( b = 0 \), or both.
• Steps to Solving: For the given example, set each factor \( n \), \( n+1 \), and \( n-6 \) equal to zero, then solve them separately resulting in solutions \( n=0 \), \( n=-1 \), and \( n=6 \).
• Simplicity and Verification: After finding potential solutions, substitute back into the original equation to verify they satisfy the equation.

By understanding and applying the correct properties, you can efficiently solve equations involving polynomials.