Problem 22 - Precalculus : Building Concepts and Connections

Question

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. \(p(x)=x^{4}-16\) (Hint: Factor first as a difference of squares.)

Step-by-Step Solution

Answer

The zeros of the polynomial are -2, 2, 2i, and -2i. The polynomial can be expressed as a product of linear factors as: \(p(x) = (x+2)(x-2)(x - 2i)(x + 2i)\)

Step 1: Factoring into difference of squares

A difference of squares formula, written as \(a^2 - b^2 = (a+b)(a-b)\), can be used to factor the given polynomial. Using this formula, the polynomial is thus factored into \(p(x) = (x^2 + 4)(x^2 - 4)\).

Step 2: Factoring further

The factor \(x^2 - 4\) can be further factored using the difference of squares formula leading to:\(x^2 - 4 = (x+2)(x-2)\). But \(x^2 + 4\) can't be factored with real coefficients and is left as is.

Step 3: Solving for values of x

Now, each factor can be set equal to zero to solve for the values of x. This gives: \(x+2=0, x-2 = 0\) and \(x^2 + 4 = 0\). Solving the first two equations gives \(x=-2\) and \(x=2\). The third equation is solved using the quadratic formula, giving: \(x= \pm 2i\). Hence, the zeros of the polynomial are -2, 2, 2i, and -2i.

Step 4: Expressing polynomial as product of linear factors

After finding the zeros of the polynomial, it can be expressed as a product of linear factors: \(p(x) = (x+2)(x-2)(x - 2i)(x + 2i).\n

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that allows for the simplification of expressions and solving of equations. It involves breaking down a polynomial into its simplest components, which are called factors. These factors are usually polynomials of lower degrees, and when multiplied together, they give back the original polynomial.

Take, for example, the polynomial in our exercise, where we start with a fourth-degree polynomial: \( p(x) = x^{4} - 16 \). The goal is to express this polynomial as a product of its factors. The first step often involves identifying patterns that can simplify the process, such as recognizing the expression as a difference of squares, which leads to the simplified form: \( p(x) = (x^{2} + 4)(x^{2} - 4) \).

The process does not end here, however; it's important to continue factoring until no further factoring is possible with real coefficients. Thus, polynomial factoring involves recognizing various factorable forms such as common factors, the difference of squares, sum and difference of cubes, and trinomials.

Difference of Squares

The difference of squares is a specific case of factoring that applies when a polynomial is expressed as the subtraction of two perfect squares. The general form is \( a^2 - b^2 = (a + b)(a - b) \). This form comes in handy in our exercise where the polynomial \( x^4 - 16 \) is recognized as a difference of squares, as both \( x^4 \) and \( 16 \) are perfect squares.

Once recognized, the polynomial is decomposed into two binomials: \( (x^2 + 4) \) and \( (x^2 - 4) \). But the factoring does not stop there, since \( (x^2 - 4) \) is itself a difference of squares and can thus be further split into the factors \( (x + 2) \) and \( (x - 2) \). It's important to know that when factoring a polynomial, you should check for the difference of squares repeatedly until all possibilities have been exhausted. However, some expressions like \( x^2 + 4 \) cannot be factored over the set of real numbers, which leads us to the concept of complex numbers.

Complex Numbers

Complex numbers extend the idea of the traditional number system composed of real numbers, by including the imaginary unit \( i \), defined by the property that \( i^2 = -1 \). With complex numbers, equations that have no real solutions, such as \( x^2 = -4 \), become solvable.

In the exercise, the polynomial \( x^2 + 4 = 0 \) cannot be factored into real numbers. However, by allowing complex solutions, we can find zeros in the form of \( x = \text{pm} 2i \), which are not real. Hence, the complete set of zeros for our polynomial includes both real zeros, -2 and 2, and non-real zeros, -2i and 2i. These zeros reflect the points where the polynomial equals zero, and their discovery is a significant aspect of solving polynomial equations.

Having both real and non-real zeros is common in polynomials of degree higher than two, and recognizing the role complex numbers play is critical for a complete understanding of polynomial solutions. This illustrates how the realms of real and complex numbers interact in the context of polynomial functions.