Problem 37 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The polynomial with real coefficients that has the given zeros is \(x^3 - 5x^2 + 9x - 5\).

Step 1: Express the polynomial

Recall that if \(a\) is a zero or root of a polynomial, then \((x-a)\) is a factor of the polynomial. Therefore, if our roots are \(1\), \(2+i\), and \(2-i\), we have the factors as \((x - 1)\), \((x - 2 - i)\), and \((x - 2 + i)\). These three factors multiply together to form the polynomial we are looking for.

Step 2: Multiply the two complex factors

Start by multiplying the two complex factors together: \((x - 2 - i)\) and \((x - 2 + i)\). When multiplied, these factors form a difference of squares: \((x - 2)^2 - i^2\). Remember that \(i^2 = -1\). Therefore, \((x - 2)^2 - (-1) = (x - 2)^2 + 1 = x^2 - 4x + 5\).

Step 3: Multiply with the real factor

Finally, multiply the resulting quadratic expression with the real factor \((x - 1)\). This will give us \((x - 1)(x^2 - 4x + 5) = x^3 - 5x^2 + 9x - 5\). This expression represents the polynomial we were looking for.

Key Concepts

Real Coefficients

When constructing polynomials with given zeros, it is crucial to recognize what 'real coefficients' means. The coefficients in a polynomial are the numbers that multiply the variables. For instance, in the polynomial P(x) = 3x^2 - 2x + 1, the coefficients are 3, -2, and 1. A real coefficient is simply a number that is not imaginary, meaning it doesn't involve the square root of a negative number, such as the imaginary unit i.

The importance of real coefficients becomes clear when working with complex zeros. In a polynomial with real coefficients, complex roots always come in pairs known as conjugate pairs. This means that if 2 + i is a zero, 2 - i, its conjugate, must also be a zero to ensure all the coefficients remain real after the multiplication of polynomials is carried out. This concept is especially relevant in the fields of algebra and electrical engineering, where the behavior of circuits is often described by polynomials with real coefficients.

Complex Roots

Complex roots are solutions to polynomial equations that involve the imaginary unit i, where i^2 = -1. They commonly appear in the form a + bi or a - bi, where a and b are real numbers. In the context of the exercise, we encounter the complex roots 2 + i and 2 - i.

Understanding complex roots is essential because they influence the shape and nature of the polynomial. Since complex numbers include both a real part (the a value) and an imaginary part (the b value), they expand the possibilities of solutions beyond the real number line. Complex roots are inherently linked with oscillatory behavior observed in nature, like wave patterns and vibrations, often modeled by polynomials in physics and engineering.

Multiplication of Polynomials

The multiplication of polynomials is a fundamental operation in algebra where two or more polynomials are multiplied together to form a single polynomial. The process involves distributing each term of one polynomial to every term of the other, often visualized using the FOIL (First, Outer, Inner, Last) method for binomials or a vertical alignment method for larger polynomials.

In our exercise, the multiplication is particularly interesting when we multiply complex conjugate pairs. The product of (x - (2 + i)) and (x - (2 - i)) is a special case known as the difference of squares, which simplifies to (x - 2)^2 + 1. During multiplication, the i terms cancel out, leaving a polynomial with real coefficients as anticipated. This property is powerfully critical in fields like signal processing and control systems where polynomial multiplication is used to analyze and design systems.