Problem 30 - College Algebra

Question

In Exercises \(29-36,\) 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. \(n=3 ; 4\) and \(2 i\) are zeros; \(f(-1)=-50\)

Step-by-Step Solution

Answer

To find the polynomial function, first find the factor form from the zeros. Then, substitute the extra point to find the leading coefficient. Finally, write out the polynomial function.

Step 1: Understand the Problem

Given are the zeros (x-values where the polynomial function equals to zero) of a third degree polynomial. These are: \(4, 2i, -2i\) (since if \(2i\) is a zero, \(-2i\) is also a zero due to polynomial function with real coefficients has conjugate pairs of complex roots). Additionally, a point \((-1, -50)\) on the function is given that can be used to determine the leading coefficient of the polynomial.

Step 2: Write the factor form of the polynomial

Use the zeros to write the factor form of the polynomial as \(f(x) = a(x-4)(x-2i)(x+2i)\)

Step 3: Simplify the Polynomial

Simplify the factors \(f(x) = a(x-4)(x^2 + 4)\).

Step 4: Determine the leading coefficient

Substitute the given point \((-1, -50)\) into the function. Like this: \(-50 = a(-1 - 4)((-1)^2 + 4)\), solve for \(a\).

Step 5: Write the Final Polynomial Function

Substitute the value of \(a\) back into the polynomial to find the final function. Check the function with the given conditions to ensure the accuracy of the solution.

Key Concepts

Real Coefficients

Polynomials with real coefficients are equations where all the coefficients are real numbers. This means every term in the polynomial is multiplied by a real number. It's important to understand this because it affects the nature of the roots of the polynomial. If a polynomial has real coefficients and includes complex numbers as roots, those complex numbers must occur in pairs—their complex conjugate pairs. Real coefficients ensure that every result from the polynomial calculations remains within the real number system, which helps maintain consistency in solving real-world problems.
This focus on real numbers allows for the polynomial to have a smooth and continuous graph along the real axis. You won't have any unexpected jumps or imaginary breaks in such polynomials' graphs when dealing with real number inputs. Real coefficients in polynomials ensure stability in various applications, from physics to engineering, where real-number solutions are required.

Complex Roots

Complex roots of polynomials are roots that include imaginary numbers. Imaginary numbers are multiples of 'i', the imaginary unit, defined as the square root of -1. Complex numbers take the form of 'a + bi', where 'a' and 'b' are real numbers.
Understanding complex roots is vital when you are analyzing polynomials with real coefficients. When you encounter a complex root, it typically signifies two roots for a polynomial: these are called conjugate pairs. If your polynomial contains a complex root '2i', it must also contain '-2i'.

• Complex roots come in pairs when the polynomial has real coefficients.
• These roots are conjugate pairs, meaning they mirror each other across the real axis.
• They do not appear as single roots unless the coefficients of the polynomial are complex as well.

Complex roots ensure that polynomials can be more flexible in their solutions, expanding the nature of the problem-solving process to include non-real numbers.

Conjugate Pairs

Conjugate pairs are an essential part of polynomials with real coefficients since they guarantee that complex roots occur in identifiable pairs. When you have a complex root like '2i', its conjugate pair would be '-2i'. This pairing is significant because it allows polynomials with real coefficients to still have roots that include complex numbers while keeping the polynomial's expression simple and real when expanded.
Imagine the polynomial expressed as \((x - 2i)(x + 2i)\). When expanded, it simplifies to \((x^2 + 4)\), a real expression. This means the imaginary components cancel out, leaving only real coefficients, thus maintaining the polynomial's properties:

• Conjugate pairs help polynomials with real coefficients remain real when expanded.
• The multiplication of conjugate pairs results in a quadratic expression in real numbers.
• It ensures symmetry in the polynomial’s root solutions, important for solving equations accurately.

Conjugate pairs help preserve the integrity of polynomial solutions in both real and complex environments.

Leading Coefficient

The leading coefficient is an important aspect of a polynomial as it is the coefficient of the term with the highest power of the variable. In polynomial equations, such as quadratic or cubic ones, the leading coefficient impacts the overall shape and orientation of its graph.
For determining the pattern and position of the polynomial's graph accurately, knowing the leading coefficient is crucial. When given a specific point like \((-1, -50)\), this can be used to find the leading coefficient 'a' in the polynomial equation. This value dictates whether the graph opens upwards or downwards and how steep it is:

• If the leading coefficient is positive, the ends of the graph go in the same upward direction.
• If negative, it typically means the graph opens downwards.
• Its magnitude affects the narrowness or wideness of the graph.

Understanding the leading coefficient allows one to predict and verify the polynomial’s behavior across its domain, aligning the function with the conditions specified in exercises and real-world scenarios alike.