Problem 33 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(5(d+8)(d-12)(d+9)=0\)

Step-by-Step Solution

Answer

The solutions to the given equation are \(d = -8, d = 12, d = -9\)

Step 1: Identify factors in the equation

Notice how the equation is formed by a multiplication of several factors \(5, (d+8), (d-12), (d+9)\) equal to zero.

Step 2: Set each factor to zero

According to the principle mentioned in the analysis, if the product is zero then at least one of these factors must be zero. Hence, set each factor equal to zero to solve for \(d\). In equation form, this means solving the following three equations separately: \(d+8=0, d-12=0, d+9=0\).

Step 3: Solve each equation

For the given equations, subtract 8 from both sides in the first equation yielding \(d = -8\). Add 12 to both sides in the second equation yielding \(d = 12\). Lastly, subtract 9 from both sides in third equation yielding \(d = -9\) .

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental concept in algebra that helps us solve polynomial equations. It states that if a product of several terms equals zero, then at least one of these terms must be zero. This is because zero is the only number that when multiplied by another number results in zero.
For example, in the equation \(5(d+8)(d-12)(d+9)=0\), we see a product of multiple factors. By applying the Zero Product Property, we know that at least one of these factors must be zero for the entire product to be zero.
This property is incredibly useful when solving polynomial equations as it allows us to break down a complex equation into simpler parts. By solving each factor individually when set to zero, we can find the possible solutions to the equation.

Factoring

Factoring is the process of breaking down a polynomial equation into simpler, multiplicative components, or 'factors', that when multiplied together, give back the original equation. As seen in our example, the equation \(5(d+8)(d-12)(d+9)=0\) is already in factored form.
Each term within the parentheses is an individual factor: \((d+8)\), \((d-12)\), and \((d+9)\). These factors show us how the polynomial can be split, simplifying the process of finding solutions.
Factoring can involve several methods including finding common factors, using the difference of squares, or employing special factoring formulas for quadratic equations. Regardless of the method, the goal is to simplify the equation to use the Zero Product Property effectively.
Recognizing when and how to factor an equation is crucial for solving polynomial equations efficiently.

Solving Equations

Solving equations involves finding the values of variables that satisfy the equation. In our original exercise, after using the Zero Product Property and factoring, each factor is set equal to zero and solved individually.
For instance:

• For \(d+8=0\), we subtract 8 from both sides to get \(d = -8\).
• For \(d-12=0\), we add 12 to both sides, giving \(d = 12\).
• For \(d+9=0\), subtract 9 from both sides to find \(d = -9\).

Each solution corresponds to a different value of \(d\) that satisfies the original equation.
By solving each part separately and understanding the relationship between the factors and their respective solutions, we can master solving any polynomial equation given in a similar form. This method makes the process manageable and systematic.