Problem 65 - 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. $$f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22$$

Step-by-Step Solution

Answer

The real zeros of the function can be found using a graphing utility, and they are a and b. The imaginary zeros are found by applying the Conjugate Pair theorem, resulting in a+bi, a-bi, b+ci and b-ci.

Step 1: Draw the graph

Plot the function \(f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22\) using a graphing utility. Find where the function intersects the x-axis. These are the real zeros of the function.

Step 2: Identify Real Zeros

From the graph, identify the real zeros. Let's say, these are a and b.

Step 3: Apply Conjugate Pair Theorem

Using the Conjugate Pair Theorem, you can find the imaginary zeros of the function. The theorem states if a+bi is a zero of a function, then so is a-bi. Because a and b are the real zeros here, you can find the other zeros using this theorem.

Step 4: Calculate Imaginary Zeros

Based on the nature of the fourth degree polynomial, there should be four zeros in total. Since a and b are the two real zeros, the remaining are imaginary zeros. Apply the Conjugate Pair theorem to calculate the imaginary zeros for a and b individually, which results in imaginary zeros a+bi, a-bi, b+ci and b-ci

Key Concepts

Graphing Functions

When dealing with functions like the polynomial function given in the exercise, graphing can be a powerful tool to visualize and understand the function's behavior. By graphing the polynomial \(f(x) = x^{4} + 3x^{3} - 5x^{2} - 21x + 22\), one can locate the points where the curve crosses the x-axis. These intersections are known as the real zeros of the function. To graph a polynomial function:

• Start by plotting the function using a graphing utility.
• Observe where the graph touches or crosses the x-axis.
• These crossing points indicate the real zeros of the polynomial, which are the values of \(x\) that satisfy \(f(x) = 0\).

After identifying these values, we can proceed to further understand and find the imaginary zeros of the function. This knowledge forms the foundation for deeper analysis using the Conjugate Pair Theorem.

Conjugate Pair Theorem

The Conjugate Pair Theorem is an essential concept when working with polynomials that have real coefficients. It is particularly useful when trying to find the zeros of polynomials with complex or imaginary roots. The theorem states that if a polynomial has real coefficients, then any non-real complex zeros must occur in conjugate pairs.
This means that if \(a + bi\) is a zero of a polynomial (where \(a\) and \(b\) are real numbers and \(b eq 0\)), then its conjugate \(a - bi\) must also be a zero.

• This concept helps us identify and confirm the imaginary zeros after finding the real zeros.
• If two of the zeros are real, as in our exercise case, the remaining two zeros must be a pair of complex conjugates.

Thus, if a polynomial of degree four has two real zeros like our example, the other two zeros will be imaginary, and you can use the theorem to determine precisely what these zeros are.

Imaginary Zeros

Imaginary zeros are an intriguing aspect of polynomial functions that don't cross the x-axis because they do not exist on the real number line. To find them, especially in higher-degree polynomials like the fourth-degree polynomial in the exercise, it's necessary to delve beyond real numbers.
Once real zeros are identified by graphing, imaginary zeros need to be calculated considering the nature and degree of the polynomial.

• Since our polynomial is of degree four, it must have four zeros in total.
• Given two real zeros previously found, the remaining are imaginary zeros, lying off the real axis.
• Utilize the Conjugate Pair Theorem to ensure these zeros occur in conjugate pairs, such as \(a+bi\) and \(a-bi\).

By employing both graphical and theoretical tools like the Conjugate Pair Theorem, we can thoroughly uncover and calculate all zeros of a polynomial function, filling in the puzzle pieces of its complete set of solutions.