Problem 81 - Algebra and Trigonometry

Question

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises 79–82. Then determine the number of real zeros and the number of imaginary zeros for each function. $$ f(x)=3 x^{4}+4 x^{3}-7 x^{2}-2 x-3 $$

Step-by-Step Solution

Answer

The specific number of real and imaginary zeros will depend on the graphing of the function \( f(x)=3x^{4}+4x^{3}-7x^{2}-2x-3 \). Generally, all polynomial functions of degree 4 have 4 zeros. The number of real zeros can be found by identifying where the graph intersects the x-axis, while the number of complex or imaginary roots is calculated by subtracting the number of real zeros from the total of 4.

Step 1: Graph the function

It should be used a graphing utility to draw the graph of \( f(x)=3x^{4}+4x^{3}-7x^{2}-2x-3 \). Observe where the graph intersects the x-axis.

Step 2: Determine the real zeros

The real zeros of the function are the x-coordinates where the graph intersects the x-axis. Identify those points if they exist.

Step 3: Determine the imaginary zeros

Subtract the number of real zeros from total degree of the polynomial. According to the Fundamental Theorem of Algebra, the result will be the number of complex or imaginary zeros.

Key Concepts

Graphing Polynomial Functions

Understanding how to graph polynomial functions is a fundamental skill in algebra that allows us to visually interpret the behavior of the function. When graphing a polynomial function, begin with identifying its degree, which is the highest exponent of the variable. The degree gives us a clue about the function's end behavior and the maximum number of turns the graph can have.

For the polynomial function f(x)=3x^{4}+4x^{3}-7x^{2}-2x-3, it is a fourth-degree polynomial, suggesting up to three turns. Using a graphing utility, you'll notice how the function behaves at extreme values of x, observing the end behaviors; it rises on both ends since the leading coefficient is positive.

The graph will also help to visually find the x-intercepts, which are crucial for determining the real zeros of the function. These x-intercepts where the graph crosses the x-axis reveal where the function's output is zero. Making an accurate graph helps in understanding how the function behaves and leads to a better grasp of its zeros.

Real Zeros of Polynomials

Real zeros of a polynomial are the values of x for which the polynomial evaluates to zero. They are also referred to as roots or solutions of the polynomial equation. When graphing a polynomial, you can identify the real zeros as the points where the graph intersects the x-axis.

From the graph of the polynomial f(x)=3x^{4}+4x^{3}-7x^{2}-2x-3, determine these intersection points. These real zeros can be identified by the x-coordinates of these points. If a graph touches the x-axis but does not cross it, it indicates a repeated zero or a zero with an even multiplicity.

It is beneficial to consider possible rational zeros using the Rational Root Theorem, which can be useful for further factoring the polynomial or using synthetic division to find all the real zeros if a graphing utility is not available.

Imaginary Zeros of Polynomials

Imaginary zeros of polynomials occur when the value for x produces a negative number under a square root, which is not possible on the real number line. These non-real zeros come in conjugate pairs, meaning if a + bi is a zero, then a - bi is also a zero where i is the imaginary unit.

After identifying real zeros of the polynomial f(x)=3x^{4}+4x^{3}-7x^{2}-2x-3 by observing the graph's x-intercepts, we can infer the presence of imaginary zeros. If the polynomial is of degree n, and there are m real zeros found, then there will be n-m imaginary zeros, as the total number of zeros must equal the degree of the polynomial.

Finding the exact imaginary zeros may require further algebraic techniques such as synthetic division, factoring, or applying the quadratic formula to the remaining factors if they are quadratic in nature.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. This theorem is the guarantee that polynomial equations will have as many solutions as their degree, although some or all of the solutions may not be real numbers.

Applied to our example, f(x)=3x^{4}+4x^{3}-7x^{2}-2x-3, we can predict that there will be four zeros in total, combining both real and imaginary zeros. After finding the number of real zeros by graphing, the remaining zeros must be imaginary. For instance, if there are two real zeros found when graphing, then by the Fundamental Theorem of Algebra, there must be two imaginary zeros for a fourth-degree polynomial.

This theorem is crucial for algebra students as it provides a framework for understanding the behavior of polynomials and ensures that when seeking solutions, they know to look for the correct total number of zeros, adjusting their search for real or complex solutions as needed.