Problem 3 - A Graphical Approach to College Algebra

Question

Find a cubic polynomial in standard form with real coefficients, having the given zeros. Let the leading coefficient be \(1 .\) Do not use a calculator. 5 and \(i\)

Step-by-Step Solution

Answer

The polynomial is \( x^3 - 5x^2 + x - 5 \).

Step 1: Understanding the Zeros

The zeros of the polynomial are given as 5 and \( i \). Since the coefficients need to be real, the complex zeros will always come in conjugate pairs. This means, if \( i \) is a zero, \( -i \) must also be a zero. As a result, the polynomial has zeros \( 5, i, \) and \( -i \).

Step 2: Construct Factor Form

A polynomial can be constructed using its zeros. If \( a \) is a zero of the polynomial, then \( (x - a) \) is a factor. Therefore, the factors of our polynomial are \((x - 5)\), \((x - i)\), and \((x + i)\).

Step 3: Multiply the Complex Factors

First, let's multiply the complex factors \((x - i)\) and \((x + i)\). Using the formula for the difference of squares, we have:\[(x - i)(x + i) = x^2 - i^2 = x^2 - (-1) = x^2 + 1 \].

Step 4: Form the Cubic Polynomial

Now, multiply the result from Step 3 with the remaining factor, \((x - 5)\):\[ (x^2 + 1)(x - 5) \].

Step 5: Expand the Expression

Distribute \(x - 5\) across \(x^2 + 1\):- First, \(x^2 \cdot x = x^3\)- Second, \(x^2 \cdot (-5) = -5x^2\)- Third, \(1 \cdot x = x\)- Lastly, \(1 \cdot (-5) = -5\).Combining all these terms, we get:\[ x^3 - 5x^2 + x - 5 \].

Step 6: Write Polynomial in Standard Form

The polynomial in standard form, with leading coefficient 1, is:\[ x^3 - 5x^2 + x - 5 \].

Key Concepts

Real Coefficients

When dealing with cubic polynomials, real coefficients are very important. They ensure that the polynomial takes real number values for real inputs. Polynomials with real coefficients can have zeros that are either real or complex. However, if they have complex zeros, those zeros come in conjugate pairs. For instance, in this exercise, the given zero is \( i \). To maintain real coefficients, the polynomial must include \(-i\) as another zero, creating the pair \( i \) and \(-i \).

This pairing is crucial because it allows any imaginary parts to cancel each other out, preserving the polynomials' real nature. In summary, real coefficients keep the polynomial relatable to real-world problems and solutions.

Complex Zeros

Complex zeros are solutions to a polynomial equation, characterized by their imaginary component. Here, one of the given zeros is \( i \). Complex numbers like this are written in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).

Due to the nature of real coefficients, complex zeros must come as conjugate pairs, meaning if \( i \) is a zero, then \(-i \) must be too. This concept helps to ensure that the polynomial maintains real coefficients, as conjugate pairs eliminate the imaginary components when multiplied. With zeros \( i \) and \(-i \), even though they are not real, they are crucial for building a polynomial with the specific type of structure required.

Factorization

Factorization involves expressing a polynomial as a product of simpler polynomials. Using the zeros of the polynomial, you can write the polynomial in its factor form. For this problem, with zeros 5, \( i \), and \(-i \), the factorization is

• \((x - 5)\)
• \((x - i)\)
• \((x + i)\)

This step is essential for constructing the polynomial. Once the factors are established, multiplication is used to compile these into a single polynomial. This step also simplifies calculations, as working with factors often makes complete expressions easier to visualize and manipulate.

Standard Form

The standard form for a polynomial is arranging the terms in descending order of their exponents. For a cubic polynomial, this means starting with the term containing \(x^3\), followed by \(x^2\), then \(x\), and finally the constant. In the example, we started with the factors

• \((x^2 + 1)\)
• \((x - 5)\)

Expanding these resulted in: \( x^3 - 5x^2 + x - 5 \). This illustrates the typical structure required.

Writing in standard form is beneficial as it provides a consistent way to present polynomials, making it easier to compare and analyze different polynomials across various calculations and areas of study.