Problem 28 - Algebra and Trigonometry

Question

In Exercises 25–32, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. $$ n=3 ; 6 \text { and }-5+2 i \text { are zeros; } f(2)=-636 $$

Step-by-Step Solution

Answer

The 3rd-degree polynomial function with real coefficients satisfying the given conditions is \( f(x) = 2x^3 + 8x^2 - 82x - 348 \).

Step 1: Set up the polynomial function

To start solving the problem, let's use the given zeros to set up the general form of the polynomial function. Since a polynomial of degree n has n roots (or zeros), we know that the polynomial function can be written in the form \( f(x) = c(x - r_1)(x - r_2)(x - r_3) \) where r1, r2 and r3 are the roots of the polynomial and c is a real constant. In this case, r1 is 6, and r2 and r3 should be -5 + 2i and -5 - 2i, as complex zeros always appear in conjugate pairs when coefficients of polynomial are real numbers.

Step 2: Express the polynomial in expanded form

Expand the function \( f(x) = c(x - 6)(x + 5 - 2i)(x + 5 + 2i) \). By using the formula \( (a + b)(a - b) = a^2 - b^2 \), this simplifies to \( f(x) = c(x - 6)(x^2 + 10x + 29) \), or further simplified to \( f(x) = c(x^3 + 4x^2 - 41x - 174) \).

Step 3: Find the coefficient c

Now, use the condition that f(2) = -636 to solve for the coefficient c. Substitute x = 2 into \( f(x) = c(x^3 + 4x^2 - 41x - 174) \) to obtain \( -636 = c(2^3 + 4*2^2 - 41*2 - 174) \). Solving the equation gives us that c = 2.

Step 4: Write down the final polynomial function

Finally, substitute c = 2 back into the equation \( f(x) = c(x^3 + 4x^2 - 41x - 174) \) we get the final polynomial function \( f(x) = 2x^3 + 8x^2 - 82x - 348 \).

Key Concepts

Complex Conjugates

Understanding complex conjugates is key when dealing with polynomial functions that have real coefficients. Complex numbers are in the form of a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit satisfying i² = -1. A complex conjugate is simply the mirror of a complex number across the real axis in the complex plane.
For a given complex number (-5 + 2i), its conjugate is (-5 - 2i). In polynomials with real coefficients, complex roots must appear in conjugate pairs. This means for every complex root,

• if (a + bi) is a root, then (a - bi) must also be a root.

This is necessary to ensure that all coefficients in the expanded form remain real. Remember, when given a complex zero, its conjugate is automatically a zero if the polynomial has real coefficients.

Roots and Zeros

The terms 'roots' and 'zeros' are often used interchangeably and refer to the same concept in polynomial functions. A zero of a function is simply a solution to the equation f(x) = 0.
In a polynomial of degree n, there are n zeros, and these zeros can be real or complex.

• For example, the polynomial given in this exercise has zeros at 6, -5 + 2i, and -5 - 2i.

These zeros can be plugged back into the polynomial to verify it equates to zero. Verifying the zeros not only helps confirm the accuracy of calculations, but provides deeper insight as to how the polynomial behaves at different points.
The roots are the x-values where the graph intersects the x-axis, and solving for these is crucial in many applications, like physics, engineering, and finances, where intersection points signify solutions to problems.

Degree of a Polynomial

The degree of a polynomial is a fundamental characteristic that determines its basic properties. The degree is the highest power of the variable x in the polynomial equation.

• For a cubic function like in the exercise, the degree is 3.

This tells us that:

• there are exactly three roots (or zeros).
• the graph of the function will have certain shapes, like turning points and end behavior dictated by this degree.

In addition, the degree influences other behavior such as the maximum number of turning points, which is always one less than the degree, and the end behavior, which is dictated by whether the degree is odd or even. Knowing this provides a comprehensive understanding of the polynomial's behavior and helps in graph plotting.

Real Coefficients

Real coefficients mean that all the numbers that serve as multipliers for the terms of a polynomial are real numbers. Real coefficients in polynomial functions ensure certain symmetrical relationships.
Understanding this concept is vital, especially when dealing with complex roots. When a polynomial has real coefficients, any complex zeros must occur in conjugate pairs. This is because the product and sum of these pairs yield real numbers,

• contributing to maintaining the real nature of the polynomial's coefficients.

For the polynomial given in the exercise, this means the terms are constructed such that combinations of complex numbers simplify into real coefficients upon expansion.
This result is crucial for ensuring the polynomial function behaves predictably in real-number contexts, making these functions usable in a wide range of practical applications.