Problem 46 - Elementary Algebra

Question

For the following problems, solve for the indicated variable. $$ x^{2}-z^{2}=0, \text { for } z $$

Step-by-Step Solution

Answer

Question: Given the equation \(x^2 - z^2 = 0\), find the possible values of \(z\). Answer: The solutions for the variable \(z\) that satisfy the given equation are \(z = -x\) and \(z = x\).

Step 1: Factor the equation

First, we need to factor the equation as a difference of squares. Recall that the difference of squares can be factored as \((a+b)(a-b)\), where \(a = x\) and \(b = z\) in our case. So, our factored equation becomes: $$(x+z)(x-z) = 0$$

Step 2: Solve for z using the Zero Product Property

Next, we utilize the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, we have two possible solutions: 1. \((x+z) = 0\) 2. \((x-z) = 0\)

Step 3: Solve each case for z

Now, we will solve each case separately: 1. \((x+z) = 0\): Subtract \(x\) from both sides: $$z = -x$$ 2. \((x-z) = 0\): Add \(z\) to both sides: $$x = z$$ So, we have two possible solutions for the variable \(z\): 1. \(z = -x\) 2. \(z = x\) In conclusion, the values of \(z\) that satisfy the given equation are \(z = -x\) and \(z = x\).

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in elementary algebra. It involves rewriting a polynomial expression as a product of simpler expressions, known as factors. This process helps to simplify equations, making them easier to work with, especially when solving for unknown variables. When you factor a polynomial, you look for common factors shared by each term in the expression and rewrite the expression as a product of these factors.

• For example, the polynomial \( x^2 - z^2 \) is a special case because it can be factored as a product of binomials due to its structure, the difference of squares.

Understanding how to factor a polynomial can help quickly solve equations, identify zeros, and understand properties of functions. And when dealing with polynomials, factoring is often the first step towards finding solutions.

Difference of Squares

The difference of squares is a specific type of polynomial that takes the form \( a^2 - b^2 \). This structure is unique because it can always be factored into the product of two binomials: \((a + b)(a - b)\). For example, in the exercise \( x^2 - z^2 = 0 \), this perfectly fits the difference of squares pattern. Here:

• \( a = x \)
• \( b = z \)

This gives us the factored form: \((x + z)(x - z)\). The difference of squares is a powerful tool because it transforms a polynomial into a simpler product form, which is crucial for further operations like solving equations. By recognizing and applying the difference of squares, students can tackle a wide range of problems with confidence and efficiency. This method significantly simplifies various algebraic expressions and problems.

Zero Product Property

The zero product property is a fundamental law in algebra that states if the product of two numbers (or expressions) is zero, at least one of the numbers (or expressions) must be zero. In mathematical terms, if \( ab = 0 \), then \( a = 0 \) or \( b = 0 \) (or both). This principle is invaluable for solving equations, especially when dealing with polynomials that have been factored, like the equation from our exercise, \((x + z)(x - z) = 0\). Using the zero product property, we can set each factor equal to zero:

• \( x + z = 0 \)
• \( x - z = 0 \)

Solving these simple linear equations gives the solutions for \( z \) as \( z = -x \) and \( z = x \). This property plays a key role in algebra by allowing students to break down complex polynomial equations into simpler, solvable parts that can yield multiple solutions. Understanding and employing this property enhances problem-solving skills and builds a solid foundation in algebra.