Problem 19 - Algebra and Trigonometry

Question

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. \(Q(x)=x^{2}+2 x+2\)

Step-by-Step Solution

Answer

\(-1 \pm i\); each zero has multiplicity 1.

Step 1: Recognize the Quadratic

The given polynomial is a quadratic in the form of \(Q(x) = x^2 + 2x + 2\). To factor it, we should first determine if it's factorable using integer factors.

Step 2: Determine the Discriminant

The discriminant of a quadratic \(ax^2 + bx + c\) is given by \(b^2 - 4ac\). In this case, \(a = 1\), \(b = 2\), and \(c = 2\). Calculate the discriminant: \(b^2 - 4ac = 2^2 - 4\times 1 \times 2 = 4 - 8 = -4\).

Step 3: Analyze the Discriminant

A negative discriminant indicates the quadratic has no real roots, so \(Q(x)\) is not factorable using real numbers. Instead, it has complex roots.

Step 4: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) to find the zeros of the polynomial. Plug in the values: \(x = \frac{-2 \pm \sqrt{-4}}{2}\).

Step 5: Simplify the Zeros

Simplify \(x = \frac{-2 \pm \sqrt{-4}}{2}\) to find the zeros. This becomes \(x = \frac{-2 \pm 2i}{2}\), which further simplifies to \(x = -1 \pm i\).

Step 6: State the Zeros and Their Multiplicity

The zeros of the polynomial are \(-1 + i\) and \(-1 - i\), each with a multiplicity of 1.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial where the highest power of the variable, often denoted as \(x\), is 2. This means that it has the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Quadratic equations can describe parabolic graphs, which means they have a distinct U shape. Understanding their structure is key to solving them not just for real roots, but also when complex numbers are involved. In this type of polynomial, solving for \(x\) means finding the values that make the equation equal zero, which are commonly referred to as the roots or zeros of the equation. These roots can appear as:

• Real and different
• Real and equal, which indicates the roots have multiplicity greater than one
• Complex, implying the roots cannot be observed on a normal graph

Knowing the form and possible characteristics of a quadratic is foundational when identifying the correct method to solve it.

Discriminant

The discriminant is a specific part of the quadratic equation defined within the quadratic formula, given by the expression \(b^2 - 4ac\). This value is crucial because it tells us about the nature of the roots without actually solving the equation fully.
Here’s a breakdown of what the discriminant reveals:

• If \(b^2 - 4ac > 0\), the quadratic has two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, technically called a double root or a root with multiplicity of 2.
• If \(b^2 - 4ac < 0\), the quadratic has two complex roots.

The given problem presented a negative discriminant \(-4\), indicating complex roots. The discriminant acts as a quick reference that decides which solution path to follow, thus saving time when determining if factoring is even possible with real numbers.

Complex Roots

Complex roots emerge from quadratic equations when the discriminant is negative. They are called complex if they have both a real and an imaginary part, often represented simply as \(a + bi\) and \(a - bi\). The presence of complex roots means the solutions aren't visible on the real number line, which changes the way we visualize or interpret such equations. In our example, \(x^2 + 2x + 2\) can't intersect the x-axis on a typical graph, as it showcases the roots \(-1 + i\) and \(-1 - i\).
Understanding complex numbers means grasping that \(i\) (the imaginary unit) is defined by \(i^2 = -1\). Complex roots often occur in conjugate pairs, which is precisely what we have in this exercise. Such elements inform us about the polynomial's behavior and are particularly common in higher mathematics, where complex functions and equations are a regular occurrence.

Quadratic Formula

The quadratic formula is a powerful tool to find the roots of any quadratic equation, written as \(ax^2 + bx + c = 0\). The formula is expressed as \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]. It's derived from completing the square of the general quadratic equation, which ensures its universality.
The formula’s versatility is due to its capability to handle any type of discriminant:

• Positive, offering real roots
• Zero, giving a single real root
• Negative, which returns complex roots

In the case provided, the quadratic formula simplifies the complex root extraction process from the quadratic \(x^2 + 2x + 2\). Plugging the values into this formula provided the roots \(-1 + i\) and \(-1 - i\), readily showing its efficiency, especially with a negative discriminant. Being comfortable using this formula is fundamental for solving quadratic equations.