Finding The Zeros Of X² - 100 A Step-by-Step Solution

Introduction: Understanding Polynomial Zeros

In the realm of mathematics, particularly within algebra, polynomial zeros, also known as roots, hold a fundamental significance. They represent the values of the variable that, when substituted into the polynomial equation, make the equation equal to zero. These zeros provide valuable insights into the behavior and characteristics of the polynomial function, serving as crucial reference points for graphing, solving equations, and understanding the function's overall nature. Identifying these zeros is not merely an academic exercise; it's a cornerstone of various mathematical applications, including curve sketching, solving optimization problems, and analyzing real-world phenomena modeled by polynomials.

For the specific polynomial presented, x² - 100, the task at hand is to determine the values of 'x' that satisfy the equation x² - 100 = 0. This polynomial, a quadratic expression, exemplifies a common type encountered in algebra. The process of finding its zeros involves employing algebraic techniques to isolate 'x' and unveil the values that nullify the expression. This exploration delves into the core principles of polynomial algebra and highlights the practical application of these principles in identifying critical points of a function. Understanding how to solve for zeros is a foundational skill that transcends the classroom, influencing fields like engineering, physics, and economics where polynomial models are frequently used to represent real-world scenarios. The subsequent sections will meticulously dissect the polynomial x² - 100, revealing the methods used to pinpoint its zeros and underscoring the broader implications of this concept in mathematics and beyond.

Deconstructing the Polynomial: x² - 100

The polynomial in question, x² - 100, is a classic example of a quadratic expression, specifically a difference of squares. This form holds a special significance in algebra due to its straightforward factorization. A quadratic expression, generally written in the form ax² + bx + c, where 'a', 'b', and 'c' are constants, represents a parabola when graphed, and its zeros correspond to the points where the parabola intersects the x-axis. In the case of x² - 100, 'a' is 1, 'b' is 0, and 'c' is -100. The absence of the 'x' term (i.e., bx) simplifies the analysis and solution process.

The expression x² - 100 is a specific instance of the difference of squares, a pattern recognized by the form a² - b². This pattern is mathematically significant because it can be factored into (a - b)(a + b). Recognizing this pattern is key to efficiently finding the zeros of the polynomial. In our case, x² corresponds to a², and 100 corresponds to b², where b is 10. Thus, the expression can be rewritten as x² - 10², making the difference of squares pattern explicitly visible. This recognition is a powerful tool in algebra, allowing for quick factorization and simplification of expressions. The ability to identify and apply the difference of squares pattern not only streamlines the process of finding zeros but also aids in solving various algebraic problems, from simplifying complex expressions to solving equations in different mathematical contexts. The next section will demonstrate how this factorization directly leads to the identification of the zeros of the polynomial, showcasing the practical utility of this algebraic identity.

Solving for Zeros: Applying the Difference of Squares

To effectively determine the zeros of the polynomial x² - 100, we leverage the difference of squares factorization. As previously established, x² - 100 can be expressed as x² - 10², which perfectly fits the pattern a² - b² = (a - b)(a + b). Applying this factorization, the polynomial transforms into (x - 10)(x + 10). This step is crucial because it converts the original quadratic expression into a product of two linear factors, making it straightforward to find the values of 'x' that make the expression equal to zero.

The zeros of a polynomial are the values of the variable that, when substituted into the polynomial, result in an output of zero. In the factored form (x - 10)(x + 10), the polynomial equals zero if either (x - 10) equals zero or (x + 10) equals zero. This principle, known as the zero-product property, is fundamental in solving polynomial equations. Setting each factor to zero independently allows us to isolate the values of 'x' that satisfy the equation. Thus, we solve two simple linear equations: x - 10 = 0 and x + 10 = 0. The solutions to these equations directly correspond to the zeros of the original polynomial.

Solving x - 10 = 0 gives x = 10, and solving x + 10 = 0 gives x = -10. These two values, 10 and -10, are the zeros of the polynomial x² - 100. They represent the points where the graph of the polynomial intersects the x-axis. This straightforward application of the difference of squares factorization, combined with the zero-product property, provides a clear and efficient method for finding the zeros of this type of quadratic polynomial. The next section will discuss the implications of these zeros and how they relate to the options provided in the original question, solidifying the understanding of the solution process.

Identifying the Correct Option: Zeros and Their Significance

Having determined the zeros of the polynomial x² - 100 to be 10 and -10, we can now pinpoint the correct option among the choices provided. The options typically present the zeros as coordinate pairs, which in this context represent the x-values that make the polynomial equal to zero. The zeros, 10 and -10, indicate that when x is 10 or -10, the expression x² - 100 evaluates to zero. These values are crucial because they define the x-intercepts of the polynomial's graph, which is a parabola in this case.

Examining the provided options, we look for the pair that includes both 10 and -10 as the zeros. Option (a) (10, 10) includes 10 but not -10. Option (b) (-10, -10) includes -10 but not 10. Option (c) (10, 0) is incorrect as it includes 0, which is not a zero of the polynomial. Option (d) (-10, 10) correctly includes both -10 and 10. Therefore, option (d) is the correct representation of the zeros of the polynomial x² - 100. This option accurately reflects the solutions we derived through the factorization and zero-product property methods.

The significance of these zeros extends beyond just solving the equation. In graphical terms, the zeros represent the points where the parabola intersects the x-axis. The symmetry of the parabola around its vertex also means that the zeros are equidistant from the axis of symmetry. In real-world applications, zeros can represent critical points in a system, such as equilibrium points in physics or break-even points in economics. Understanding how to find and interpret zeros is therefore a fundamental skill in both theoretical mathematics and practical problem-solving. The final section will summarize the steps taken to solve the problem and reinforce the key concepts learned in the process.

Conclusion: Recap and Key Takeaways

In summary, finding the zeros of the polynomial x² - 100 involved a series of algebraic techniques and a deep understanding of polynomial properties. The process began with recognizing the polynomial as a difference of squares, a specific pattern that allows for straightforward factorization. This recognition is a critical step, as it simplifies the problem significantly. The difference of squares pattern, a² - b² = (a - b)(a + b), was applied to transform x² - 100 into (x - 10)(x + 10).

Next, the zero-product property was employed. This property states that if the product of two factors is zero, then at least one of the factors must be zero. This principle allowed us to set each factor, (x - 10) and (x + 10), equal to zero independently, leading to two simple linear equations. Solving these equations, x - 10 = 0 and x + 10 = 0, yielded the zeros of the polynomial: x = 10 and x = -10. These values are the solutions to the equation x² - 100 = 0 and represent the x-intercepts of the polynomial's graph.

Finally, the zeros were matched to the provided options, and option (d), (-10, 10), was correctly identified as the answer. This process not only solves the specific problem but also reinforces broader concepts in algebra, such as factorization, the zero-product property, and the significance of polynomial zeros. Understanding these concepts is crucial for tackling more complex problems in mathematics and various applied fields. The ability to efficiently find zeros of polynomials is a foundational skill that underpins many mathematical and scientific applications, making this exercise a valuable learning experience.