Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$\frac{2}{3},-1,3+\sqrt{2} i$$

Step-by-Step Solution

Answer

One of the polynomials that has the zeros \(\frac{2}{3}, -1, 3+\sqrt{2}i\) is \(3x^4-8x^3+17x^2-10x+4\).

Step 1: Identify the Complex Conjugate

Since we know that complex roots comes in pairs when the coefficients are real, identify the complex conjugate of \(3+\sqrt{2}i\) which is \(3-\sqrt{2}i\).

Step 2: Convert Roots to Factors

Next, convert all these roots into polynomial factors. A root r corresponds to a factor of the form \(x-r\). So our roots \(\frac{2}{3}\), -1, \(3+\sqrt{2}i\), and \(3-\sqrt{2}i\) translate in the factors \(x- \frac{2}{3}\), \(x+1\), \(x-(3+\sqrt{2}i)\), \(x-(3-\sqrt{2}i)\) respectively.

Step 3: Calculate the Polynomial

Calculate the polynomial by multiplying these factors together: \((x-\frac{2}{3})(x+1)[x-(3+\sqrt{2}i)][x-(3-\sqrt{2}i)]\). Simplifying this multiplication gives the polynomial \(3x^4-8x^3+17x^2-10x+4\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. An imaginary number is a multiple of the imaginary unit denoted as \(i\), where \(i\) is defined by the property \(i^2 = -1\). So, a complex number is often expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
Understanding complex numbers is crucial because they expand the concept of one-dimensional number lines to two-dimensional planes, allowing for more comprehensive mathematical analysis and solutions.

• Real part: the real number component of a complex number.
• Imaginary part: the multiple of \(i\), representing the imaginary dimension.

For example, in the given polynomial exercise, we have the complex number root \(3 + \sqrt{2}i\). Here, 3 is the real part and \(\sqrt{2}i\) is the imaginary part.

Real Coefficients

When dealing with polynomials, having real coefficients means that the numbers in front of the variables (like \(x\) in a polynomial) are all real numbers. Real numbers include both rational and irrational numbers but do not include imaginary numbers.
When identifying a polynomial with real coefficients, it’s essential to remember that any complex roots must appear in conjugate pairs. This ensures that when multiplied out, the imaginary components cancel each other out, leaving only real numbers as coefficients.
Thus, if a polynomial has real coefficients, complex roots will contribute symmetric factors to the polynomial.

• Real coefficients imply symmetry in complex roots.
• Important for ensuring all terms in a polynomial remain within the real number system.

This is why in our original solution, the complex root \(3 + \sqrt{2}i\) necessitated including its conjugate \(3 - \sqrt{2}i\).

Roots and Zeros

In mathematics, the terms roots and zeros of a polynomial are often used interchangeably. They represent the values of \(x\) that make the polynomial equal to zero.
When we say a polynomial has a zero at \(x = r\), it means that \(f(r) = 0\), where \(f\) represents the polynomial function.

• Roots denote solutions to the equation \(f(x) = 0\).
• Each root corresponds to a factor in the form \(x - r\).

In the exercise at hand, we were given zeros \(\frac{2}{3}\), \(-1\), and the complex number \(3 + \sqrt{2}i\) along with its conjugate. Each zero translates into a corresponding factor for the polynomial, allowing us to understand its structure and polynomial expression.

Conjugates

Conjugates are pairs of expressions that resemble each other but with the sign of the imaginary part changed. When dealing with complex numbers, the conjugate of a complex number \(a + bi\) is \(a - bi\).
These are crucial in mathematics for several reasons:

• Ensuring polynomials with real coefficients.
• Facilitating the elimination of imaginary parts during multiplication.

In the context of polynomials with real coefficients, it's necessary to include the conjugate of any complex root to keep the coefficients real. This ensures balance and is an application of the complex conjugate root theorem.
For the given polynomial exercise, this meant pairing the root \(3 + \sqrt{2}i\) with its conjugate \(3 - \sqrt{2}i\) to maintain entirely real coefficients after multiplying together the polynomial factors.