Problem 15 - 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. $$p(x)=x^{2}-\pi^{2}$$

Step-by-Step Solution

Answer

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

Step 1: Find Zeros of the Polynomial

Set the polynomial equal to zero and solve for \(x\). So \(x^{2} - \pi^{2} = 0\). This equation can be factored using the difference of squares formula \((a - b)(a + b) = a^2 - b^2\).

Step 2: Applying the Difference of Squares Formula

Apply the difference of squares formula, to factor the equation. Here, \(a = x\) and \(b = \pi\). Thus, the factored form is \((x-\pi)(x+\pi)=0\).

Step 3: Solve for X

Set each factor equal to 0 and solve for \(x\). Thus, \(x-\pi=0\) gives \(x=\pi\), and \(x+\pi = 0\) gives \(x=-\pi\). These are all the zeros of the polynomial \(p(x) = x^{2} - \pi^{2}\).

Step 4: Express Polynomial as a Product of Linear Factors

A linear factor is an expression equal to zero at the root of the function. The function \(p(x) = x^{2} - \pi^{2}\) would therefore be equal to the product of linear factors: \((x - \pi)(x + \pi)\).

Key Concepts

Difference of Squares

The difference of squares is a common algebraic pattern that allows us to factor certain types of quadratic expressions easily. It's based on the formula \(a^2 - b^2 = (a + b)(a - b)\). When facing a polynomial like \(x^2 - \pi^2\), we can identify \(a\) as \(x\) and \(b\) as \(\pi\). After applying the difference of squares formula, we get \( (x + \pi)(x - \pi) \), which are the factors that can then be used to find the zeros of the polynomial.

The difference of squares is particularly useful because it can simplify complex problems by breaking them down into simpler parts, which makes solving them a process of just finding the values of \(x\) that satisfy each factor when set to zero.

Linear Factors

Linear factors are the building blocks of polynomial functions. A linear factor is a first-degree polynomial, which means it has the form \(x - c\), where \(c\) is a constant. Each zero of the polynomial corresponds to a linear factor. For instance, if \(\pi\) is a zero of \(p(x)\), then \(x - \pi\) is a linear factor of the polynomial. By identifying the zeros of the polynomial, we can express \(p(x)\) as a product of its linear factors, such as \(p(x) = (x - \pi)(x + \pi)\). This representation is valuable because it provides a straightforward depiction of the polynomial's behavior and its intersections with the \(x\)-axis.

Factoring Polynomials

Factoring polynomials is the process of decomposing a polynomial into a product of its factors, which can be simpler polynomials. The method used to factor a polynomial depends on its degree and the form of its terms. For quadratic polynomials like \(x^2 - \pi^2\), common factoring techniques include using the difference of squares, factoring by grouping, or applying special products. Factoring is a critical step in solving polynomial equations because it can transform a complex equation into simpler, solvable components. By factoring \(x^2 - \pi^2\) into \( (x + \pi)(x - \pi) \), we prepare the equation for finding its zeros.

Understanding factoring also strengthens the skill of simplifying expressions before attempting to solve an equation, which can make finding solutions more manageable.

Solving Quadratic Equations

Solving quadratic equations is a staple in algebra. A quadratic equation in the standard form looks like \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. There are various methods to solve these equations, such as factoring, using the quadratic formula, completing the square, or even graphing. In the case of the equation \(x^2 - \pi^2 = 0\), we factor it as a difference of squares and then solve by setting each factor equal to zero. This gives us the two solutions \(x = \pi\) and \(x = -\pi\), which correspond to the points where the graph of the polynomial intersects the \(x\)-axis.

Mastering solving quadratic equations not only helps with finding zeros of polynomials but also equips students with tools to analyze and understand the properties of various functions and models.