Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=x^{2}+25$$

Step-by-Step Solution

Answer

The zeros of the function \(f(x)=x^{2}+25\) are \(x = \pm 5i\). The polynomial can be written as product of linear factors as follows: \(f(x)=(x-5i)(x + 5i)\). The graph of the function reflects these findings as it does not cross the x-axis, hence verifying that there are no real roots.

Step 1: Finding zeros

To find the zeros of \(f(x)\), we need to solve the equation \(x^{2}+25=0\). Subtract 25 from both sides to get \(x^{2}=-25\). Next, take the square root of both sides. Since we can't take the square root of a negative number, we have \(x= \pm 5i\), where i is the unit of imaginary numbers.

Step 2: Write as linear factors

Next, we need to express \(f(x)\) as linear factors. A linear factor is an expression of the form \(x - a\), where a is a root of the polynomial. Since the roots of \(f(x)\) are \(\pm 5i\), then the linear factors are \(x - 5i\) and \(x + 5i\). So \(f(x)\) can be expressed as \(f(x)=(x-5i)(x + 5i)\)

Step 3: Verify with graphing utility

Last step is to graph the equation in a graphing utility to visually verify the solution. Since we have imaginary roots we can't visually confirm these zeros on the graph. The graph of a quadratic function \(f(x)=x^2+25\) is a parabola that opens upwards and is shifted 25 units upwards on the y-axis. This indicates that the function doesn't cross the x-axis and hence has no real roots which coincides with our earlier solution.

Key Concepts

Complex Roots

Complex roots arise in polynomial equations when there is no real number that satisfies the equation. As seen in the exercise, when trying to find the zeros of the polynomial function f(x) = x^2 + 25, we confront the equation x^2 = -25. Normally, a negative number under a square root leads us to the realm of imaginary numbers.

By defining the imaginary unit i as the square root of -1, we are then allowed to write the square root of -25 as 5i, where i indicates that the number is an imaginary quantity. The zeros of the polynomial are thus ±5i, which are complex roots because they can't be plotted on the regular number line. Complex roots always come in conjugate pairs, which in this case are (5i, -5i).

Linear Factors of Polynomials

In algebra, expressing a polynomial as a product of linear factors is essentially breaking down the polynomial into simpler, bite-sized pieces. These factors are of the form (x - a), where a is a zero of the polynomial. The given exercise demonstrates that once we know the zeroes of the polynomial, we can directly form the linear factors.

For instance, given the zeros ±5i for our polynomial f(x), we can write the linear factors as (x - 5i) and (x + 5i). Multiplying these factors out will result back in the original polynomial, confirming that they are indeed correct. Understanding linear factors is crucial because it simplifies many aspects of algebra, such as solving equations and graphing.

Graphing Quadratic Functions

Graphing quadratic functions is a fundamental skill when studying algebra and calculus. These functions typically have the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola. Whether the parabola opens upwards or downwards depends on the coefficient a.

As for the polynomial in our exercise, f(x) = x^2 + 25, its graph is a parabola that opens upwards (since a = 1, which is positive) and does not intersect with the x-axis, indicating there are no real roots. Indeed, verifying with a graphing utility will show that the vertex of this parabola is at the point (0,25) on the y-axis. This graphical representation supports our algebraic finding that the zeros are complex.

Imaginary Numbers

Imaginary numbers are an essential concept in mathematics, particularly when studying complex roots. They are defined as multiples of i, where i is the imaginary unit with the special property that i^2 = -1.

Students often find imaginary numbers perplexing because they cannot be represented on the traditional number line. However, they are vital in extending our understanding of numbers and are used in various fields including engineering and physics. For example, in the provided exercise, the zeros of the polynomial were found to be ±5i. These are purely imaginary numbers, meaning they have no real number component. It is important to grasp the concept of imaginary numbers as they are fundamental to the study of complex numbers and polynomial functions with complex roots.