Problem 66 - 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. $$r^{3}=81 r$$

Step-by-Step Solution

Answer

The short answer to the given equation \(r^3 = 81r\) is that the possible values for r that satisfy the equation are \(r = 0\), \(r = -9\), and \(r = 9\).

Step 1: Rewrite the Equation

To make the equation easier to factor, rewrite it by subtracting 81r from both sides: \[r^3 - 81r = 0\]

Step 2: Factor the Equation

Now, we need to factor the equation. We can first factor out r to simplify the expression: \[r(r^2 - 81) = 0\] Next, we can see that the term inside the parentheses (\(r^2 - 81\)) is a difference of squares (since \(81 = 9^2\)). We can factor this as follows: \[r(r + 9)(r - 9) = 0\]

Step 3: Apply the Zero Product Rule

Now that we have factored the equation, we can apply the zero product rule. This rule states that if a product of factors is equal to zero, then at least one of the factors must be equal to zero. In this case, we have three factors: \(r\), \(r + 9\), and \(r - 9\). Therefore, either \(r = 0\), \(r + 9 = 0\), or \(r - 9 = 0\). Solving these equations for r gives us the possible values for r: r = 0 r + 9 = 0 => r = -9 r - 9 = 0 => r = 9 So, the possible values for r that satisfy the given equation are \(\boldsymbol{r = 0}\), \(\boldsymbol{r = -9}\), and \(\boldsymbol{r = 9}\).

Key Concepts

Zero Product Rule

The Zero Product Rule is an essential tool when solving polynomial equations that have been factored. This rule tells us that if you multiply two or more numbers together and the result is zero, then at least one of the numbers must be zero. This is because zero is the only number that can "absorb" any number when multiplied by it. Therefore, if a product of several terms is zero, one or more of these terms must be zero to satisfy the equation.

In algebra, we use this rule to solve equations once they've been factored into simpler terms. When you come across a factored equation, such as

• \[(r)(r+9)(r-9) = 0\]

You take each factor and set it equal to zero. Thus, you would set each of the following equal to zero:

• \(r = 0\)
• \(r + 9 = 0\)
• \(r - 9 = 0\)

Solving each of these separate equations gives you all the possible solutions for the original polynomial equation. In this case, these solutions would be

• \(r = 0\)
• \(r = -9\)
• \(r = 9\)

These are the values for \(r\) where the original equation equals zero.

Difference of Squares

The Difference of Squares is a special factoring pattern used in algebra. It applies when you have two perfect squares separated by a subtraction sign. The formula is

• \(a^2 - b^2 = (a + b)(a - b)\)

This pattern is quite handy when factoring polynomial expressions.

In our exercise, the term

• \(r^2 - 81\)

can be factored as a difference of squares. Here,

• \(a = r\)
• \(b = 9\) (since \(81 = 9^2\))

Applying the difference of squares formula results in

• \((r + 9)(r - 9)\)

This allows you to simplify and break down expressions that might seem complex at first glance. Recognizing this pattern can save time and effort in solving various algebraic problems, especially when they involve quadratic-like scenarios in a polynomial context.

Solving Polynomial Equations

When solving polynomial equations, we often use a combination of strategies, such as factoring and special algebraic rules. These equations are expressions involving multiple powers of a variable. Our goal is to find the values of the variable that make the equation true.

The first step in our example was to rewrite the equation

• \(r^3 - 81r = 0\)

By moving all terms to one side, it becomes easier to factorize. This equation presents a cubic polynomial. We start by factoring out the common term, in this case,

• \(r\). This results in
  • \(r(r^2 - 81)\)

Recognition of the second expression

• \((r^2 - 81)\) as a difference of squares allowed us to factor further.

Once fully factored, the Zero Product Rule is applied to each factor, determining where at least one factor equals zero.

Different types of polynomials will require different strategies, like recognizing patterns such as quadratics or difference of squares. Practice makes it easier to know which path to take, whether it's straightforward factoring, applying rules like the Zero Product Rule, or covering other algebraic techniques. The end goal remains the same: finding all possible solutions that satisfy the polynomial equation.