Problem 65 - Beginning and Intermediate Algebra

Question

The following equations are not quadratic but can be solved by factoring and applying the zero product rule. Solve each equation. $$m^{3}=64 m$$

Step-by-Step Solution

Answer

The given equation is \(m^3 = 64m\). First, rewrite the equation as \(m^3 - 64m = 0\). Then, factor out the common term to get \(m(m^2 - 64) = 0\). Now, factor the difference of squares to obtain \(m(m+8)(m-8)=0\). Applying the zero-product rule, the solutions are \(m = 0\), \(m = -8\), and \(m = 8\).

Step 1: Write down the equation

We are given the equation: \(m^3 = 64m\)

Step 2: Move all terms to one side

Subtract \(64m\) from both sides of the equation: \(m^3 - 64m = 0\)

Step 3: Factor out the common term (GCF)

Notice that both terms have a common factor of \(m\). Factor it out: \(m(m^2 - 64) = 0\)

Step 4: Factor the remaining quadratic term

Now we have a difference of squares inside the parentheses. Factor it as well: \(m(m+8)(m-8)=0\)

Step 5: Apply the zero product rule

Now, use the zero product rule, which states that if the product of factors is zero, then at least one of those factors must be equal to zero. So we have three possibilities: 1. \(m=0\) 2. \(m+8=0\) 3. \(m-8=0\)

Step 6: Solve for m

For each case, solve for \(m\): 1. \(m=0\) 2. \(m=-8\) 3. \(m=8\) So the solutions to the given equation are \(m = 0\), \(m = -8\), and \(m = 8\).

Key Concepts

Zero Product Rule

The zero product rule is a fundamental principle in algebra that comes in handy when solving equations that have been factored. It states that if the product of two or more numbers (or expressions) is zero, then at least one of the factors must be zero. This rule is essential because it allows us to solve equations by setting each factor to zero and solving for the variable.

For example, in the equation we arrive at after factoring,

• \( m(m+8)(m-8) = 0 \)

we apply the zero product rule by setting each individual factor equal to zero:

• \( m = 0 \)
• \( m + 8 = 0 \)
• \( m - 8 = 0 \)

We then solve these simple equations to find all possible solutions: \( m = 0 \), \( m = -8 \), and \( m = 8 \).
So, whenever you can break an equation down into multiple factors, the zero product rule is your best friend to find all potential values for the variable.

Polynomial Equations

Polynomial equations involve expressions composed of variables raised to whole number powers and their coefficients. They're versatile and can be presented in many forms. The equation from our example,

• \( m^3 = 64m \)

is a polynomial equation since it involves variable \( m \) raised to powers. Moving all terms to one side,

• \( m^3 - 64m = 0 \)

turns it into a polynomial equation ready for factoring.

Solving polynomial equations often involves:

• Factoring to break down the equation into simpler binomial or monomial factors.
• Using algebraic techniques like simplifying expressions, and applying the zero product rule.

The goal is to make the equation easier to handle by finding its roots, which are the solutions satisfying the equation set to zero. Thus, understanding how to manipulate polynomial equations is crucial for solving them efficiently.

Difference of Squares

The difference of squares is a special factoring technique useful for expressions in the form \( a^2 - b^2 \). This expression can be factored into

• \((a + b)(a - b)\)

making it easier to solve or further simplify. This pattern is crucial when you have terms with squared variables subtracting from each other.
In our solution, after factoring out the greatest common factor, we reached

• \( m^2 - 64 \)

as a part of the equation. Recognizing it as a difference of squares since \(64\) is \(8^2\), we factored it as:

• \( (m + 8)(m - 8) \)

This factoring allows for the straightforward application of the zero product rule. Mastery of this technique is very helpful in solving polynomial equations, as it simplifies the expressions you have to work with.

Greatest Common Factor (GCF)

When solving equations through factoring, identifying the greatest common factor (GCF) is often the first crucial step. This factor is the largest quantity that divides all terms of the expression, helping simplify the equation by factoring it out. For the equation

• \( m^3 - 64m = 0 \)

we notice that \( m \) is common in both terms. Factoring out \( m \), we get

• \( m(m^2 - 64) = 0 \)

This simplifies the equation and prepares it for further factoring or simplification.

Here's why finding the GCF is critical:

• It reduces the complexity of polynomial expressions by decreasing their degree.
• It avoids unnecessary complexity in solving the equation.
• Ensures that no potential solutions are overlooked.

By making the equation simpler, you can focus on solving the reduced polynomial, bringing the factors closer to applying the zero product rule efficiently.