Problem 31 - 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. \(n=4 ;-2,5,\) and \(3+2 i\) are zeros; \(f(1)=-96\)

Step-by-Step Solution

Answer

The 4th degree polynomial function is \(f(x) = 2(x + 2)(x - 5)(x^2 -6x + 5)\)

Step 1: Identify all zeros

The given zeros of the polynomial are \(-2\), \(5\) and \(3+2i\). Since the coefficients of the polynomial are real, the zeros of the polynomial come in conjugate pairs. Hence, the conjugate of \(3+2i\), which is \(3-2i\), is also a zero of the polynomial. So, we have 4 zeros in total, corresponding to the degree of the polynomial.

Step 2: Write the polynomial using the zeros

The polynomial can be expressed as:\[f(x) = a(x - (-2))(x - 5)(x - (3+2i))(x - (3-2i))\]This simplifies to:\[f(x) = a(x + 2)(x - 5)(x - 3 - 2i)(x - 3 + 2i)\]\[f(x) = a(x + 2)(x - 5)((x - 3)^2 - (2i)^2)\]\[f(x) = a(x + 2)(x - 5)(x^2 -6x + 9 - 4)\]\[f(x) = a(x + 2)(x - 5)(x^2 -6x + 5)\]

Step 3: Use the given function value to find the leading coefficient

Substitute \(x=1\) and \(f(1) =-96\) into the equation:\[-96 = a(1 + 2)(1 - 5)((1 - 3)^2 + 5)\]\[-96 = a(-6)(-2)(4)\]Solving for \(a\), we get \(a = 2\).

Key Concepts

Zeros of a Polynomial

The zeros of a polynomial are the values of \(x\) for which the polynomial evaluates to zero. These zeros are also known as the roots of the polynomial. Finding zeros is essential because they give us the solutions to polynomial equations. In this context, the zeros are given as \(-2\), \(5\), and \(3 + 2i\). It is important to note that these zeros play a crucial role in forming the polynomial function.

When identifying zeros, remember:

• If the polynomial has real coefficients, any complex zero must have its complex conjugate also as a zero.
• This means if \(3 + 2i\) is a zero, its conjugate \(3 - 2i\) must also be a zero.

Together, these zeros end up determining the structure and characteristics of the polynomial.

Complex Conjugates

Complex conjugate pairs are crucial when dealing with polynomials that have real coefficients. A complex number is often represented in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit \(\sqrt{-1}\). The conjugate of this complex number is \(a - bi\).

Here's why complex conjugates matter:

• When a polynomial has real coefficients, complex zeros must occur in conjugate pairs to ensure that the polynomial remains real-valued.
• This ensures that the polynomial function will still intercept the real-axis at various points even with complex components.

In our example with the zero \(3 + 2i\), the conjugate \(3 - 2i\) guarantees the polynomial has real coefficients without imaginary components in the result.

Degree of a Polynomial

The degree of a polynomial is determined by the highest power of \(x\) in the polynomial expression. It is an essential factor as it tells us the maximum number of roots (or zeros) the polynomial will have. In this scenario, the degree \(n\) is 4.

Aspects to consider about the degree of polynomials:

• The degree gives insight into the graph of the function, indicating the possible turning points and the end behavior as \(x\) approaches infinity or negative infinity.
• For a fourth-degree polynomial, we have four zeros, including both real and complex zeros. Given that complex zeros appear in conjugate pairs, this count offers a check on our setup.

Thus, the degree is a fundamental property that shapes the polynomial's behavior and complexity.

Real Coefficients

In polynomials, having real coefficients means all the numbers multiplying the variables are real numbers. This is significant because it dictates certain properties of the polynomial, especially concerning zeros.

Why real coefficients are crucial:

• The property of having real coefficients affects how complex roots are managed, requiring complex zeros to appear in conjugate pairs (like \(3 + 2i\) and \(3 - 2i\)) so that the imaginary parts cancel out, maintaining real values.
• This impacts the construct and form of the polynomial, influencing both how it can be written and graphed.

Understanding and using real coefficients helps ensure that polynomials can easily be analyzed and applied in real-world scenarios, where only real values make sense.