Problem 63 - Algebra 2

Question

Find a fourth-degree polynomial function with zeros \(1,-1, i,\) and \(-i .\) Write the function in both factored form and standard form.

Step-by-Step Solution

Answer

The polynomial in factored form is \( (x - 1)(x + 1)(x - i)(x + i) \), and in standard form, it is \( x^4 - 1 \).

Step 1: Identifying the Zeros

The given zeros of the polynomial are 1, -1, i, and -i. Since complex zeros occur in conjugate pairs, the zeros i and -i confirm that our polynomial will have real coefficients.

Step 2: Writing the Factors

If a polynomial has zeros at x = a, then it has factors of the form (x - a). Thus, given our zeros, the polynomial in factored form will have factors (x - 1)(x + 1)(x - i)(x + i).

Step 3: Multiplying the Complex Conjugate Factors

Multiply the factors with complex zeros (x - i)(x + i) to find a real quadratic factor. Since they are conjugates, their product is \((x - i)(x + i) = x^2 - i^2 = x^2 + 1\).

Step 4: Multiplying the Real Factors

Multiply the remaining factors with real zeros (x - 1)(x + 1) to find a real quadratic factor. Using the difference of squares, \((x - 1)(x + 1) = x^2 - 1\).

Step 5: Combining Quadratic Factors

Multiply the two quadratic factors to obtain the polynomial in standard form: \((x^2 + 1)(x^2 - 1)\).

Step 6: Expanding to Standard Form

Expand the product of the quadratic factors: \((x^2 + 1)(x^2 - 1) = x^4 - x^2 + x^2 - 1 = x^4 - 1\), which represents the polynomial in standard form.

Key Concepts

Complex Conjugate Zeros

When dealing with polynomial functions, one intriguing property is the occurrence of complex conjugate zeros. These come in pairs that are mirror images of each other with respect to the real axis on the complex plane. In other words, if a polynomial with real coefficients has a complex zero like \( a + bi \), it must also have \( a - bi \) as a zero. This stems from the Fundamental Theorem of Algebra, which ensures that non-real zeros of real polynomials always come in conjugate pairs.
As an example, the conjugate pairs \( i \) and \( -i \) from our exercise are such that their product will always yield a real number, specifically the product is \( i^2 = -1 \), hence multiplying \( (x - i)(x + i) = x^2 - i^2 = x^2 + 1 \), a real quadratic factor. This knowledge is crucial because it simplifies the process of finding a polynomial's factored form when complex zeros are present, ensuring that the coefficients of the polynomial remain real numbers.

Factored Form of Polynomials

The factored form of a polynomial is an expression that breaks down the polynomial into its simplest multipliers, which are its zeros. Each zero of the polynomial, represented as \( x = a \), translates to a factor of \( (x - a) \). The beauty of this form is that it makes the zeros of the polynomial readily apparent.
For instance, with the fourth-degree polynomial in our exercise, we can see at a glance its zeros: 1, -1, \( i \), and \( -i \), when expressed in factored form as \( (x - 1)(x + 1)(x - i)(x + i) \). This form is particularly advantageous when solving polynomial equations because it allows us to directly apply the Zero Product Property, which tells us that if a product of factors equals zero, at least one of the factors must be zero.

Standard Form of Polynomials

The standard form of a polynomial is another way to write down the same mathematical expression, typically arranging the terms in descending powers of \( x \) and combining like terms. The highest power of \( x \) indicates the polynomial’s degree. This form is highly beneficial for visualizing the general shape of the graph of the polynomial and understanding the behavior of the function as \( x \) approaches positive or negative infinity.
In our exercise, we multiplied the factors related to the polynomial's zeros to transition from the factored form to the standard form. The expanded form of our given polynomial, \( x^4 - 1 \), clearly reveals that it is a fourth-degree polynomial, as the highest power of \( x \) is 4.

Polynomial Zeros

The zeros of a polynomial, also known as roots or solutions, are the values of \( x \) that make the polynomial equal to zero. Finding the zeros is a fundamental aspect of solving polynomials because they provide valuable insights, such as the x-intercepts of the graph and the potential turning points of the function. Moreover, the zeros help in sketching the graph and in many cases, in determining the polynomial’s end behavior when combined with its degree.
By utilizing the given zeros: 1, -1, \( i \), and \( -i \), and recognizing their implications, we have elegantly crafted a fourth-degree polynomial function for the exercise. These zeros are instrumental in constructing the polynomial's factored form which eventually leads us to the standard form after expanding the product of the factors.