Problem 33 - 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. \(n=4 ; i\) and \(3 i\) are zeros; \(f(-1)=20\)

Step-by-Step Solution

Answer

The nth-degree polynomial function that satisfies the given conditions is \(f(x)=x^4+10x^2+9\).

Step 1: Recognize conjugate zeros

If a polynomial function has real coefficients, then complex zeros must occur in conjugate pairs. If \(i\) and \(3i\) are zeros, their conjugates \(-i\) and \(-3i\) are also zeros of the function.

Step 2: Write the polynomial in factored form

Using the fact that zeros of the function give the factors of the polynomial, one can write the polynomial function in its factored form: \(f(x)=a(x-i)(x+i)(x-3i)(x+3i)\).

Step 3: Simplify the polynomial

Multiplying the factors yields the polynomial function: \(f(x)=a(x^2+1)(x^2+9)=a(x^4+10x^2+9)\).

Step 4: Find the scaling factor 'a'

Substitute \(x=-1\) in the equation and set \(f(x)\) to \(20\) to find the scaling factor \(a\): \(20=a((-1)^4+10(-1)^2+9)=a(20)\), which gives \(a=1\). So, the polynomial we were looking for is \(f(x)=x^4+10x^2+9\).

Step 5: Verify the answer

By substituting the zeros back to the found polynomial function and plugging in \(x=-1\), it can be verified that the polynomial found is indeed consistent with the given conditions.

Key Concepts

Complex Conjugate Zeros

Complex conjugate zeros are a fundamental concept when dealing with polynomial functions that have real coefficients. A complex number has the form \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit, satisfying the equation \(i^2 = -1\). The complex conjugate of \(a + bi\) is \(a - bi\). When a polynomial function has real coefficients, like in our exercise, and it possesses a complex zero \(a + bi\), it will always have the complex conjugate \(a - bi\) as a zero as well.

This pairing occurs because when you multiply a complex number by its conjugate, the result is a non-negative real number, helping ensure that all coefficients of the polynomial remain real. Thus, if \(i\) and \(3i\) are zeros of the given polynomial, their conjugates \( -i\) and \( -3i\) must also be zeros, maintaining the rule that a polynomial with real coefficients has conjugate complex zeros.

Factored Form of Polynomial

The factored form of a polynomial expresses the function as a product of its factors, which can be based on its zeros. One useful property of polynomials is that if \(r\) is a zero, then \(x-r\) is a factor. For a 4th-degree polynomial function with zeros \(i\), \(3i\), \( -i\), \( -3i\), we can write the function as \(f\(x\)=a\(x-i\)\(x+i\)\(x-3i\)\(x+3i\)\).

Notice how each zero has translated to a linear factor. The constant \(a\) is called the leading coefficient or scaling factor and affects the graph's vertical stretch or compression but does not affect the position of the zeros. Simplifying the multiplied factors will give us a polynomial in standard form, which can be achieved by multiplying each pair of conjugate factors, eventually leading to the expanded polynomial with real coefficients.

Real Coefficients

When we discuss real coefficients in the context of polynomials, we're referring to the numbers that multiply each term, which consist of \(x\) raised to a power, being actual real numbers. These coefficients impact the shape and location of the graph but are especially important because they maintain the polynomial's nature of being a real function.

In our example, finding the correct value of the leading coefficient \(a\) was critical to satisfy the given condition \(f(-1)=20\). This step highlights the process of determining the coefficient by using the polynomial's known values, which in our exercise, led us to conclude that \(a = 1\). With this, the polynomial reflects its behavior as a real function, ensuring that when graphed, it will cross or touch the x-axis at the defined zeros and achieve the specified function value. Furthermore, due to the real coefficients, the polynomial will possess certain symmetry and predictability in its graph as well.