Problem 23 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ r(t-6)(t+8)=0 $$

Step-by-Step Solution

Answer

\( r = 0 \), \( t = 6 \), \( t = -8 \)

Step 1: Understand the Zero Product Property

The equation is given as \( r(t-6)(t+8)=0 \). According to the zero product property, if the product of several factors equals zero, then at least one of the factors must be zero. Therefore, we'll set each factor to zero to find the possible values of \( r \) and \( t \).

Step 2: Set Each Factor to Zero

Apply the zero product property. We do this by setting each of the factors in the equation to zero separately:1. \( r = 0 \)2. \( t - 6 = 0 \)3. \( t + 8 = 0 \)

Step 3: Solve Each Factor Equation

Solving each equation:1. \( r = 0 \) is already solved.2. \( t - 6 = 0 \) leads to \( t = 6 \).3. \( t + 8 = 0 \) leads to \( t = -8 \).

Step 4: Present the Solution Set

The solutions to the equation \( r(t-6)(t+8)=0 \) are all the values that satisfy any of the factor equations. Therefore, the solution set is \( r = 0 \), \( t = 6 \), or \( t = -8 \).

Key Concepts

Solving Equations

The process of solving equations involves finding the value(s) for which the equation becomes true. In simple terms, this means finding the unknown numbers or variables that satisfy the given equation. With the equation \( r(t-6)(t+8) = 0 \), our goal is to determine the values for \( r \) and \( t \) that make the whole expression equal to zero.

To solve an equation like this, we use the Zero Product Property, a fundamental concept in algebra. This property states that if the product of two or more factors is zero, at least one of the factors must be zero. This is because multiplying any number by zero gives zero. Therefore, if we have an equation where multiple terms are multiplied together and the result is zero, we can infer valuable information by equating each factor to zero individually.

In our exercise, we have three factors: \( r \), \( t-6 \), and \( t+8 \). By setting each factor to zero, we systematically explore all possible solutions.

Polynomial Equations

Polynomial equations are mathematical expressions involving sums and products of variables and coefficients raised to whole number powers. In this exercise, the equation \( r(t-6)(t+8) = 0 \) is a polynomial equation. Here, the polynomial is expressed in a factored form consisting of three linear factors: \( r \), \( t-6 \), and \( t+8 \).

Yet, this equation is not in the typical polynomial format like \( ax^n + bx^{n-1} + \ldots + c = 0 \). Instead, it highlights the Zero Product Property that's useful in solving polynomial equations when written as a product of factors. Linear polynomial equations like \( t-6 = 0 \) and \( t+8 = 0 \) are special cases where the degree of the polynomial is 1.

Recognizing polynomial equations and understanding their characteristics empowers you to apply suitable strategies for finding solutions. Here, the strategy involves breaking down the polynomial into manageable parts and taking advantage of properties like the zero product to solve it.

Solution Set

The solution set of an equation is the collection of all values that satisfy the equation, making it true. In our specific problem, once we've set each factor equal to zero, we've identified potential solutions:

• First, \( r = 0 \) is a straightforward solution since the equation becomes zero immediately when \( r \) is zero.
• Second, solving \( t-6 = 0 \) yields \( t = 6 \), meaning that when \( t \) equals 6, the entire product becomes zero given the presence of the term \( t-6 \).
• Lastly, \( t+8 = 0 \) results in \( t = -8 \), offering another valid solution.

Hence, the complete solution set is \( r = 0 \), \( t = 6 \), or \( t = -8 \).

This set includes all the scenarios where the original polynomial equation holds true. Understanding the solution set is crucial as it tells us not only the answers but also informs us about the conditions under which the equation maintains its integrity.