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Chapter 1: Problem 13

Solve each equation by factoring. $$ 5 x^{3}-20 x=0 $$

Short Answer

Expert verified
The solutions are \(x = 0\), \(x = 2\), and \(x = -2\).

Step by step solution

01

Identify Common Factors

First, look for the greatest common factor (GCF) in the equation. Both terms, \(5x^3\) and \(-20x\), have a common factor of \(5x\). Factoring out \(5x\), we rewrite the equation: \[ 5x(x^2 - 4) = 0 \]
02

Factor the Quadratic Expression

Now, factor the quadratic expression \(x^2 - 4\). This is a difference of squares, which can be factored as:\[ x^2 - 4 = (x - 2)(x + 2) \]Substituting back, the equation becomes:\[ 5x(x - 2)(x + 2) = 0 \]
03

Apply Zero Product Property

Using the zero product property, when a product of factors equals zero, at least one of the factors must be zero. Set each factor equal to zero and solve:- \(5x = 0\) implies \(x = 0\).- \(x - 2 = 0\) implies \(x = 2\).- \(x + 2 = 0\) implies \(x = -2\).
04

Conclude the Solution

Now conclude that the solutions to the equation are the values of \(x\) found in Step 3:\(x = 0\), \(x = 2\), and \(x = -2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor
When solving equations by factoring, identifying the greatest common factor (GCF) is often the first step. The GCF is the largest factor that can evenly divide all terms in the equation. For example, in the expression \(5x^3 - 20x\), both terms have a common factor of \(5x\).
This is determined by:
  • Looking at the numerical coefficients, which are 5 and 20 in this case. They have a common factor of 5.
  • Identifying the lowest power of the variable shared by all terms. Here, both terms contain \(x\), with the lowest power being 1.
By factoring out the GCF \(5x\), we simplify the equation, making it \(5x(x^2 - 4) = 0\). This step reduces the complexity and prepares the expression for further factoring or solving.
Difference of Squares
The concept of "difference of squares" is useful in algebra for simplifying expressions. It's recognized when you have a subtraction between two perfect squares. A classic form is \(a^2 - b^2\).
This can be factored into \((a - b)(a + b)\).
In our case, the expression \(x^2 - 4\) follows this form:
  • Identify \(a^2\) as \(x^2\) and \(b^2\) as \(2^2\).
  • Substitute into the formula to get \((x - 2)(x + 2)\).
This simplification enables us to more easily solve for solutions when combined with the zero product property. Recognizing patterns like a difference of squares helps streamline algebraic processes.
Zero Product Property
The zero product property is a fundamental principle in algebra. It states that if the product of multiple factors equals zero, then at least one of the factors must be zero. This concept is pivotal when solving equations by factoring. Consider the fully factored equation \(5x(x - 2)(x + 2) = 0\). According to the zero product property:
  • If \(5x = 0\), then \(x = 0\).
  • If \(x - 2 = 0\), solve to find \(x = 2\).
  • If \(x + 2 = 0\), solve to find \(x = -2\).
The beauty of this property lies in its simplicity; it breaks down complex equations into straightforward solutions. Identifying these zeroes helps us solve for unknown variables efficiently.

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