Problem 6 - Algebra Form and Function

Question

Find the zeros of the polynomials. $$ x^{4}-1 $$

Step-by-Step Solution

Answer

Answer: The zeros of the polynomial \(x^4 - 1\) are -1, 1, -i, and i.

Step 1: Factor the polynomial

We can use the difference of squares formula to factor the polynomial. The difference of squares formula states that: $$ a^2 - b^2 = (a + b)(a - b) $$ First, recognize that \(x^4 - 1\) can be written as \(x^4 - 1^2\) which is a difference of squares. So using the formula, we have: $$ x^4 - 1 = (x^2 + 1)(x^2 - 1) $$ Now, notice that \((x^2 - 1)\) is also a difference of squares. So, we can apply the formula again: $$ x^4 - 1 = (x^2 + 1)(x + 1)(x - 1) $$

Step 2: Solve for x in each factor

Now that we have factored the polynomial, we can set each factor to zero and solve for x: $$ x^2 + 1 = 0 \\ x^2 = -1 \\ x = \pm\sqrt{-1} $$ For the other factors: $$ x + 1 = 0 \\ x = -1 \\ x - 1 = 0 \\ x = 1 $$

Step 3: Compile the zeros

We have found the zeros of the polynomial \(x^4 - 1\). They are: $$ x = -1, 1, -i, i $$ So, the zeros of the polynomial \(x^4 - 1\) are -1, 1, -i, i.

Key Concepts

Difference of Squares

The concept of the difference of squares is a powerful tool in algebra that helps us factor certain polynomials efficiently. A difference of squares occurs when you have two square numbers separated by a subtraction sign. The general formula for the difference of squares is:

• \( a^2 - b^2 = (a + b)(a - b) \)

This formula is derived from the pattern that emerges when you multiply \((a + b)\) and \((a - b)\). When expanded, the middle terms cancel out, leaving only the squares of each term. Applying this to our problem, \(x^4 - 1\) can be rewritten as \(x^4 - 1^2\), perfectly fitting the difference of squares format.
Recognizing this allows us to factor the expression as

• \((x^2 + 1)(x^2 - 1)\)

We can further factor \((x^2 - 1)\) as another difference of squares, giving us the fully factored expression

• \((x^2 + 1)(x + 1)(x - 1)\)

This technique not only simplifies expressions but also helps in solving equations, as it breaks down complex problems into simpler components.

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that involves breaking down a polynomial into simpler, multiplied subexpressions (factors) that, when expanded, return the original polynomial. In our exercise, we used factoring to find the roots of \(x^4 - 1\).
When factoring, strategies like grouping, the use of special products (like difference of squares), and trial and error for small degree polynomials are common approaches. The goal is to rewrite the polynomial as a product of lower-degree polynomials.
In the example of \(x^4 - 1\), we started by identifying it can be expressed as

• \((x^2 + 1)(x^2 - 1)\)

Following this, we further broke down \((x^2 - 1)\) into

• \((x + 1)(x - 1)\)

Each of these factors is easier to handle and solve for zero, which helps in finding the polynomial's roots. The factoring process thus greatly simplifies the task of solving polynomial equations.

Complex Numbers

Complex numbers are an extension of the real numbers and are used to solve equations that have no real solutions. They include the imaginary unit \(i\), where

• \(i = \sqrt{-1}\)

This means that \(i^2 = -1\), a property that doesn't hold for any real number. In our polynomial example, we encountered complex numbers when solving the factor \(x^2 + 1 = 0\).
Here, setting \(x^2 = -1\) wouldn't give a real number solution, as no real number squared gives a negative. Instead, we introduce the imaginary unit, yielding solutions

• \(x = \pm i\)

These solutions, \(-i\) and \(i\), are complex numbers. Understanding complex numbers involves grasping how they operate in arithmetic (addition, subtraction, multiplication, division) and how they can be represented geometrically in the complex plane.
Utilizing complex numbers broadens our capability to solve polynomial equations fully, as every polynomial equation can be factored or solved over the field of complex numbers, guaranteeing a comprehensive set of roots.