Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to find the real zeros of the function, and then (b) use the real zeros to find the exact values of the imaginary zeros. $$f(x)=25 x^{3}-55 x^{2}-54 x-18$$

Step-by-Step Solution

Answer

The real zeros can be found using a graphing utility. If the polynomial equation were to have any imaginary zeros, they could be found once the function is expressed as a factored polynomial.

Step 1: Finding Real Zeros

Firstly, you should input the function \(f(x) = 25 x^{3} - 55 x^{2} - 54 x - 18\) into a graphing utility. The 'x-intercepts' (where the function meets the x-axis) will represent the real zeros of the function. Take note of these points as they will be required in the next step.

Step 2: Factoring the polynomial

Once the real zeros are found, rewrite the equation in a factored form. For example, if the real zeros are a, b, and c, the equation can be written as \(f(x) = 25(x-a)(x-b)(x-c)\).

Step 3: Finding Imaginary Zeros

In this case, the given function is a cubic one, which means it will always have at least one real zero, so there may not be any imaginary zeros. But if there were only one or two real zeros, the other zero(s) may be complex or imaginary. These can be found by substitifying the real zeros in the polynomial and solving the quadratic equation that results.

Key Concepts

Real Zeros

Real zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. These values are also known as roots or solutions. To determine the real zeros of a polynomial function, you can use a graphing utility where the x-intercepts indicate the places where the function crosses or touches the x-axis.

For the function \(f(x) = 25 x^{3} - 55 x^{2} - 54 x - 18\), inputting it into a graphing calculator will help visually locate these points.

• The real zeros tell us that the polynomial has solutions that can be plotted on a real number line.
• They are the values where the function transitions through zero in a graph.

By understanding where these intersections with the x-axis occur, you are one step closer to determining the overall behavior of the polynomial across the real number spectrum.

Imaginary Zeros

The concept of imaginary zeros comes into play when a polynomial doesn't fully intersect the real number line. These zeros occur when the remaining solutions of a polynomial are not real numbers. Imaginary zeros usually involve the square root of negative numbers which are represented with the imaginary unit \(i\) (where \(i^2 = -1\)).

In our polynomial function, since it's a cubic polynomial, it has at least one real zero. However, if there are fewer than three real zeros for the cubic function, the rest might be imaginary. To find these, typically, factoring or using the quadratic formula can illustrate these non-real solutions.

• Imaginary zeros don't appear on the graph of the polynomial in the x-intercept format.
• These zeros often come in conjugate pairs, meaning if \(a + bi\) is a zero, \(a - bi\) is also a zero.

Imaginary zeros expand our understanding of polynomial solutions beyond real numbers and are an essential part of complex numbers.

Factoring Polynomials

Factoring a polynomial involves expressing it as a product of its individual factors. These factors are often in the form of linear, quadratic, or higher degree polynomials. Identifying the factors helps simplify the polynomial to see its roots more clearly. Factoring can be straightforward when the real zeros are known, as they can directly lead to the factors.

For example, if a polynomial has real zeros at \(x = a, b,\) and \(c\), the polynomial could be expressed as \(f(x) = (x - a)(x - b)(x - c)\) transformed into its factored form. Here:

• Each real zero corresponds to a factor of the polynomial that crosses through zero.
• If a polynomial can be divided by a polynomial of lesser degree, it is factorable.

The process of factoring breaks down the polynomial, making it easier to find all kinds of zeros, whether real or imaginary.

Quadratic Equation

The quadratic equation is a principle tool in finding zeros, especially imaginary ones, of polynomials. A standard quadratic equation is \(ax^2 + bx + c = 0\). To solve finding the zeros in quadratic format, the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is applied.

This formula provides solutions to the quadratic component revealed in polynomial factorization. The expression under the square root, \(b^2 - 4ac\), known as the discriminant, determines the nature of the zeros. If it's negative, the zeros are imaginary.

• A positive discriminant indicates two distinct real zeros.
• A zero discriminant results in exactly one real zero (repeated).
• A negative discriminant signifies two complex conjugates (imaginary zeros).

Understanding and applying the quadratic equation helps uncover zeros some may overlook, providing a deeper insight into the structure and behavior of polynomials.