Problem 21 - Algebra and Trigonometry

Question

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. \(P(x)=x^{3}+4 x\)

Step-by-Step Solution

Answer

The polynomial \( x^3 + 4x \) factors to \( x(x^2 + 4) \), with zeros at \( x = 0 \) (multiplicity 1), \( x = 2i \) (multiplicity 1), and \( x = -2i \) (multiplicity 1).

Step 1: Identify Common Factors

First, recognize that every term in the polynomial \( P(x) = x^3 + 4x \) contains the factor \( x \). So, we can factor out \( x \) from each term.\[ P(x) = x(x^2 + 4) \]

Step 2: Factor Completely

Next, observe the quadratic expression \( x^2 + 4 \) inside the parentheses. Notice that this expression is a sum of squares, which generally cannot be factored with real coefficients. Therefore, this expression is already in its simplest form over the real numbers.Thus, the complete factorization over the real numbers is:\[ P(x) = x(x^2 + 4) \]

Step 3: Find Zeros of the Polynomial

To find the zeros of the polynomial, set each factor equal to zero:1. For \( x = 0 \): This is one zero.2. For \( x^2 + 4 = 0 \): Rearrange to get \( x^2 = -4 \). Since \( x^2 = -4 \), the solutions are \( x = \pm 2i \) by taking the square root on both sides.Thus, the zeros of the polynomial are \( x = 0, x = 2i, \text{ and } x = -2i \).

Step 4: Determine the Multiplicity of Each Zero

The zero \( x = 0 \) comes from the factor \( x \), which has an exponent of 1, so \( x = 0 \) has a multiplicity of 1.The zeros \( x = 2i \) and \( x = -2i \) arise from a quadratic factor \( x^2 + 4 \), each also occurring once.So, the multiplicities are:- Zero \( x = 0 \) has multiplicity 1.- Zero \( x = 2i \) has multiplicity 1.- Zero \( x = -2i \) has multiplicity 1.

Key Concepts

Zeros of Polynomials

Finding the zeros of a polynomial is a fundamental aspect of algebra. Zeros, also known as roots or solutions, are the values of the variable that make the polynomial equal to zero. To find the zeros of any polynomial, you factorize the polynomial into simpler expressions and set each factor equal to zero. For example, given the polynomial

• \( P(x) = x^3 + 4x \), factor out the common factor: \[ P(x) = x(x^2 + 4) \]
• Set each factor to zero: \( x = 0 \), and \( x^2 + 4 = 0 \).

Finding zeros helps you understand the behavior of the graph of the polynomial function, as these zeros indicate points where the graph intersects the x-axis. Keep in mind that some zeros might not be visible on the graph if they are complex numbers. Understanding how to find and compute these zeros is crucial in solving polynomial equations effectively.

Complex Numbers

Complex numbers often come into play when solving polynomial equations, especially when a polynomial includes terms that result in negative values under a square root, such as \(x^2 + 4 = 0\). The concept of complex numbers extends the idea of numbers beyond real numbers to include imaginary elements. The imaginary unit \(i\) is defined such that \(i^2 = -1\). Therefore, solutions to equations like \(x^2 + 4 = 0\) can be expressed as:

• Rewriting the equation gives \(x^2 = -4\).
• Take the square root on both sides to find solutions: \(x = \pm 2i\).

This demonstrates how complex numbers, which include both real and imaginary parts, can be essential in accurately solving and factoring polynomials. Complex solutions indicate that the polynomial does not have all intersections with the real number line and sometimes extend our view to solutions that exist within the complex plane.

Multiplicity of Zeros

When examining polynomials, noting the multiplicity of each zero is essential. In the context of factored polynomials, multiplicity refers to the number of times a particular zero appears as a solution. Consider the polynomial \( P(x) = x(x^2 + 4) \) and its zeros:

• \(x = 0\) comes from the factor \(x\), appearing once, giving it a multiplicity of 1.
• \(x = 2i\) and \(x = -2i\) both arise from \(x^2 + 4\), appearing once each, each with multiplicity 1.

Multiplicity tells us more about the nature of the graph at each zero. A zero with even multiplicity touches the x-axis and returns, indicating a local minimum or maximum, while a zero with odd multiplicity will cross the x-axis. Understanding multiplicity is crucial for graphing polynomials and analyzing their features.