Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$5, i,-i$$

Step-by-Step Solution

Answer

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

Step 1: Find the factors of the polynomial

First, convert each zero to its corresponding factor. Specifically, the zeros 5, \(i\), and \(-i\) correspond to the factors \(x - 5\), \(x - i\), and \(x + i\) respectively.

Step 2: Multiply the factors

Then multiply all of these factors together to get the polynomial function, i.e.\((x - 5)(x - i)(x + i)\).

Step 3: Simplify the polynomial

Now, simplify the product, keeping in mind that \(i^2 = -1\), so \((x - i)(x + i) = x^2 + 1\). Thus, our polynomial is \((x - 5)(x^2 + 1)\) which simplifies to \(x^3 - 5x^2 + x - 5\).

Key Concepts

Finding Polynomial Factors

Polynomials are algebraic expressions that involve a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient. To understand polynomials better, we can look at how to find the roots or zeros of a polynomial equation, which helps us determine its factors. A 'zero' of a polynomial is a number that, when substituted into the polynomial, yields a value of zero. These zeros are crucial as they correspond directly to factors of the polynomial.

For example, if a polynomial has a zero at x = 5, we can say that \(x - 5\) is a factor of the polynomial. Finding the factors from given zeros involves reversing the process of evaluating a polynomial. For real zeros, if \(a\) is a zero, \(x - a\) is the factor. If zeros are complex, which often come in conjugate pairs, each complex zero \(a + bi\) yields two factors: \(x - (a + bi)\) and \(x - (a - bi)\).

When given a set of zeros, it's essential to remember that each zero corresponds to a factor of the polynomial. If the polynomial has real coefficients, complex zeros will always occur in conjugate pairs, such as \(i\) and \(i\). By finding the factors from the given zeros, you can reconstruct the polynomial by multiplying these factors together.

Complex Zeros of a Polynomial

When dealing with polynomials, complex zeros play an intriguing role. A complex zero typically comes in the form \(a + bi\), where \(i\) represents the imaginary unit, which is defined such that \(i^2 = -1\). According to the Fundamental Theorem of Algebra, every non-constant polynomial function with complex coefficients has at least one complex root, and if the coefficients are real, the complex roots appear in conjugate pairs.

The importance of this theorem cannot be overstated, as it assures us that a polynomial of degree \(n\) will have \(n\) roots in the complex number system (counting multiplicity). Conjugate pairs mean if \(a + bi\) is a zero of a polynomial with real coefficients, then \(a - bi\) will also be a zero. In the given exercise, the complex zeros provided are \(i\) and \(\-i\). When finding a polynomial with these zeros, the corresponding factors would be \(x - i\) and \(x + i\). Multiplying these together simplifies to \(x^2 + 1\), a real polynomial, because the product of a complex conjugate pair always results in a real-numbered polynomial expression.

Simplifying Polynomial Expressions

Simplifying polynomial expressions is a fundamental skill that allows you to reduce a polynomial to its most manageable form. This process often involves combining like terms and using algebraic identities to make the expression simpler or more useful for further manipulation or solving equations.

In the context of our exercise, after finding the factors of the polynomial \(x^2 + 1\) and \(x - 5\), you would typically multiply these to get the polynomial function. When multiplying \(x^2 + 1\) by \(x - 5\), it involves distributing each term in one binomial across the terms in the other, also known as the FOIL method (First, Outer, Inner, Last).

Simplifying such products can often result in combining like terms, and recognizing opportunities to apply special product formulas, such as squaring a binomial or multiplying conjugates. Always check if the expression can be factored after simplifying; sometimes, the simplified form reveals more information about the polynomial's properties and its graph.