Problem 23 - College Algebra

Question

In Exercises \(23-28\), factor each polynomial: a. as the product of factors that are irreducible over the rational numbers. b. as the product of factors that are irreducible over the real numbers. c. in completely factored form involving complex nonreal, or imaginary, numbers. $$ x^{4}-x^{2}-20 $$

Step-by-Step Solution

Answer

a. The polynomial is factored into \(x^{2}(x-5)(x+5)\) over the rational numbers. b. To factorize it over the real numbers, the factors are \(x^{2}(x-5)(x+5)\). And c. Over complex or non-real numbers, the factors of the polynomial become \(x(x+\sqrt{5})(x-\sqrt{5})(x)\).

Step 1: Factor the Polynomial Over the Rational Numbers

Firstly, identify rational factors. The given polynomial is of the form \(x^{4}-x^{2}-20\). For factorization over the rational numbers, it can be written in the form \(x^{2}(x^{2} - 1) -20\). The quadratic equation \(x^{2} - 1 - 20 = 0\) can be factored further into \((x-5)(x+5)\). Subsequently, the polynomial factorization over the rational numbers becomes \(x^{2}(x-5)(x+5)\).

Step 2: Factor the Polynomial Over the Real Numbers

The factorization over the real numbers will be the same factorization over the rational numbers, since every rational number is a real number. Therefore, the polynomial factorization over the real numbers stays as \(x^{2}(x-5)(x+5)\).

Step 3: Factor the Polynomial Over the Complex Numbers

Completing the complete factorization involving imaginary or complex numbers requires finding the roots of the polynomial. As mentioned in Step 2, the roots of the Polynomial are \(x = 0, -\sqrt{5}, \sqrt{5}\). Thus, the polynomial completely factored over the complex numbers can be written as \(x(x+\sqrt{5})(x-\sqrt{5})(x)\).

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. They can be positive, negative, or zero. Common examples of rational numbers include \( \frac{1}{2} \), \( -3 \), and \( 0 \). It is essential to note that every integer is also a rational number because it can be written as a fraction with a denominator of 1, such as \( 5 = \frac{5}{1} \).

• Properties: Rational numbers adhere to properties of basic operations like addition, subtraction, multiplication, and division (except division by zero).
• Fractions: They can be simplified, multiplied, added, or subtracted, and they have specific decimal representations. Some are terminating (e.g., \( 0.75 \)), while others are repeating (e.g., \( 0.333... \)).

Understanding rational numbers helps in polynomial factorization because when we say "factor over the rational numbers," we seek factors that can be expressed with these numbers. In the polynomial \( x^4-x^2-20 \), factorization over the rational numbers, such as breaking it down to \( x^2(x-5)(x+5) \), uses integers which are indeed rational numbers.

Real Numbers

Real numbers include all the numbers on the number line. This means all the rational numbers plus all the irrational numbers (which cannot be expressed as simple fractions, such as \( \pi \) and \( \sqrt{2} \)). Real numbers encompass both positive and negative numbers, as well as zero. They are used to measure continuous quantities and are essential in geometry and calculus.

• Completeness: The real numbers fill up the entire number line without any gaps. This is the least property, meaning every nonempty subset of real numbers that is bounded above has a least upper bound in real numbers.
• Dense: Between any two real numbers, there exists another real number. This ensures the richness and continuity of the real number line.

For polynomial factorization, real numbers are crucial because factoring over the real numbers (such as \( x^2(x-5)(x+5) \)) means we include any real solution that does not necessarily have to be rational, although in most contexts involving basic polynomial equations, rational solutions often suffice.

Complex Numbers

Complex numbers extend the idea of one-dimensional quantities like rational and real numbers to two dimensions. They consist of a real part and an imaginary part, written as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).

• Operations: Similar to real numbers, they can be added, subtracted, multiplied, and divided, while taking into account the property of \( i \).
• Representation: They can be represented on a plane called the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

Polynomial equations often require complex numbers for complete solution sets. For example, the polynomial \( x^4-x^2-20 \) in its completely factored form using complex numbers, is expressed using roots such as \( x(x+\sqrt{5})(x-\sqrt{5}) \). This provides insight into all possible roots, thus fully solving the polynomial beyond real number solutions.

Imaginary Numbers

Imaginary numbers are a component of complex numbers and arise from taking the square root of negative numbers. The notation \( i \) is used to denote the imaginary unit, where \( i = \sqrt{-1} \). Imaginary numbers are not "real" in the sense of physics or physical counting, but they are essential in engineering and science.

• Basic Property: The square of the imaginary unit, \( i^2 \), is defined as \(-1\). This allows computation with negative square roots.
• Application: Imaginary numbers, when combined with real numbers, form complex numbers (e.g., \( 3 + 4i \)), which can represent real-world phenomena like oscillations in electrical engineering.

In polynomial factorization, the introduction of imaginary numbers can help in cases where real roots do not exist. This allows for a complete factorization over complex numbers, capturing all solutions and providing deeper insight into the characteristics of polynomial equations.