Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Confirm that the function has the indicated zeros. $$f(x)=x^{2}+5 ;-\sqrt{5} i, \sqrt{5} i$$

Step-by-Step Solution

Answer

-\sqrt{5}i and \sqrt{5}i are indeed the roots of the function \(f(x)=x^{2}+5\).

Step 1: Understand the Problem

We are given that the roots of function \(f(x)=x^{2} + 5\) are \(-\sqrt{5}i\) and \(\sqrt{5}i\). We have to verify this.

Step 2: Substitution of the first root

We substitute the first root \(-\sqrt{5}i\) into the function. Hence, \(f(-\sqrt{5}i) = (-\sqrt{5}i)^{2} + 5\). This simplifies to \(f(-\sqrt{5}i) = -5 + 5 = 0\), proving that \(-\sqrt{5}i\) is indeed a root of the function.

Step 3: Substitution of the second root

Next, we substitute the second root \(\sqrt{5}i\) into the function. Hence, \(f(\sqrt{5}i) = (\sqrt{5}i)^{2} + 5\). This simplifies to \(f(\sqrt{5}i) = -5 + 5 = 0\), which further proves that \(\sqrt{5}i\) is also a root of the function.

Key Concepts

Polynomial Roots

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. The roots, or zeros, of a polynomial function are the values of the variable that make the function equal to zero. Identifying these roots is a fundamental aspect of solving polynomial equations and understanding the function's behavior.

For a simple polynomial, such as a quadratic function, the roots can often be found using the quadratic formula or factoring. However, higher-degree polynomials might require more complex methods like synthetic division, the Rational Root Theorem, or numerical approximations. Each real root represents an x-intercept on the graph of the polynomial function. The existence of complex roots, reveals that not all intercepts are necessarily visible on the standard Cartesian coordinate plane.

Finding the roots is not just an academic exercise; it has practical applications in various fields such as physics, engineering, and economics where the roots represent critical points in a system modeled by a polynomial equation.

Imaginary Numbers

In mathematics, imaginary numbers are a sophisticated concept that extends the real number system. An imaginary number is defined as a multiple of the imaginary unit i, where i is the square root of -1. This might seem abstract at first, because in the system of real numbers there is no number that when squared gives a negative result.

The inclusion of imaginary numbers allows for the development of a broader set of numbers called complex numbers, denoted as a + bi where a and b are real numbers. Complex numbers enable us to solve polynomial equations that have no real-number solutions. This is particularly useful in situations involving quadratic functions, where the discriminant is negative, indicating the absence of real roots. The concept helps us to visualize and work with phenomena that can't be represented on a one-dimensional number line, expanding our reach in both theoretical and applied mathematics.

Root Verification

The process of root verification is critical to ensure that the solutions proposed for a polynomial equation are correct. Verifying a root is fundamentally straightforward: one substitutes the potential root into the polynomial equation and checks to see if the left side of the equation equals zero. If after the substitution, the equation holds true (equals zero), then the value is indeed a root.

Practical Application of Root Verification

From an educational standpoint, students should always carry out root verification to confirm their results. In a real-world context, this due diligence prevents errors in calculations that could lead to faulty designs, incorrect scientific models, or financial miscalculations. When dealing with complex roots, it's important to remember to substitute the complex conjugates of the roots into the equation as well, as polynomials with real coefficients will have complex roots occurring in conjugate pairs.