Problem 18 - 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. $$p(x)=x^{2}-5$$

Step-by-Step Solution

Answer

The zeros of the polynomial \(x^{2} - 5\) are \(\sqrt{5}\) and \(-\sqrt{5}\). The polynomial can be expressed as a product of linear factors: \(p(x) = (x - \sqrt{5})(x + \sqrt{5})\).

Step 1: Find the Zeros

Start by setting the polynomial equal to 0 to find the zeros. Thus, the equation will look as follows: \(x^{2} - 5 = 0\).

Step 2: Solve the Equation

Solving the equation involves finding the x-values that make the equation equals zero. Add 5 to both sides to isolate \(x^{2}\) on the left, then find the square root of both sides. The result is \(x = \sqrt{5}\) and \(x = -\sqrt{5}\). This means that \(\sqrt{5}\) and \(-\sqrt{5}\) are the zeros of the polynomial.

Step 3: Express the Polynomial as a Product of Linear Factors

Expressing the polynomial as a product of linear factors means writing the polynomial as a multiplication of linear expressions. Based on Step 2, the polynomial can be expressed as follows: \(p(x) = (x - \sqrt{5})(x + \sqrt{5})\).

Key Concepts

Quadratic Equations

A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero. These equations are called 'quadratic' because the variable \(x\) is raised to the power of 2, which is also known as the degree of the polynomial. Solving a quadratic equation typically involves finding the values of \(x\) that satisfy the equation, known as the roots or zeros of the equation.

For example, in the problem \(x^2 - 5 = 0\), we have a quadratic equation where \(a=1\), \(b=0\), and \(c=-5\). Since there is no \(bx\) term, it simplifies the equation, allowing us to solve it by taking the square root of both sides.

There are various methods to solve quadratic equations such as factoring, using the quadratic formula, or completing the square. In cases where the equation simplifies easily, as with \(x^2 = 5\), direct manipulation using square roots is a straightforward approach.

Linear Factors

Linear factors are expressions of the form \(ax + b\), where \(a\) and \(b\) are constants and \(a\) is not zero. When a polynomial is expressed as a product of linear factors, it means that the polynomial can be broken down into multiple linear terms multiplied together. This is an important concept because it gives us a way to represent polynomials in their simplest factored form.

In the problem, we found that the zeros of the polynomial \(x^2 - 5\) are \(\sqrt{5}\) and \(-\sqrt{5}\). From these zeros, we can write the polynomial as a product of linear factors: \(p(x) = (x - \sqrt{5})(x + \sqrt{5})\).

This factored form is useful for a number of reasons. It simplifies solving polynomial equations, helps in understanding the behavior of the graph of the polynomial, and is useful in calculus for finding intercepts and analyzing functions. By representing a polynomial as a product of linear factors, we gain insight into its roots and the locations where the polynomial crosses the x-axis.

Polynomial Equations

Polynomial equations are equations involving polynomial expressions, which consist of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. These equations vary in degree, which is the highest power of the variable in the polynomial. A quadratic equation is a type of polynomial equation with a degree of 2.

The equation \(x^2 - 5 = 0\) from the problem is a simple polynomial equation of degree 2. Polynomial equations can often be expressed in various forms. One important form is the factored form, which expresses the polynomial as a product of its linear factors. This is particularly useful for identifying zeros or solutions to the equation.

Polynomial equations serve as the foundation for a wide range of mathematical analysis and applications. They appear in many fields such as physics, engineering, and economics. Understanding how to solve polynomial equations and express them in factored form is essential for deeper mathematical study, including calculus, where analyzing the behavior of polynomial functions is key.