Problem 91 - Precalculus : Building Concepts and Connections

Question

In this problem, you will explore the relationship between factoring a quadratic expression over the complex numbers and finding the zeros of the associated quadratic function. This topic will be explained in greater detail in Chapter \(3 .\) (a) Multiply \((x+i)(x-i)\) (b) What are the zeros of \(f(x)=x^{2}+1 ?\) (c) What is the relationship between your answers to parts (a) and (b)? (d) Using your answers to parts (a)-(c) as a guide, how would you factor \(x^{2}+9 ?\) (e) Using you answers to parts (a)-(d) as a guide, how would you factor \(x^{2}+c^{2},\) where \(c\) is a positive real number?

Step-by-Step Solution

Answer

A quadratic function \(f(x) = x^{2} + c^{2}\), where \(c\) is a real positive number, can be factored into \((x+ci)(x-ci)\). The zeros of the function are \(i*c\) and \(-i*c\).

Step 1: Multiply \((x+i)(x-i)\)

This step involves simplifying the expression \((x+i)(x-i)\). The easiest way is to use the identity \(a^{2} - b^{2} = (a+b)(a-b)\) with \(a = x\) and \(b = i\). This results in \(x^{2} - i^{2}\). Because by definition, \(i^{2} = -1\), it simplifies to \(x^{2} +1\).

Step 2: Find the zeros of \(f(x) = x^{2} + 1\)

The zero(s) of a function are the x-value(s) where the function equals zero. Suppose \(x^{2} + 1 = 0\), moving \(1\) to the other side, and taking square root on both sides, we have \(x = \pm i\). So the zeros of the function are \(+i\) and \(-i\).

Step 3: Explain the relationship

The factors of the expression \(x^{2} + 1\) are \((x+i)\) and \((x-i)\), which are related to the zeros of the function \(f(x) = x^{2} + 1\). In general, for a quadratic equation \(a(x-h)^{2} = 0\), \(h\) is the zero of the function. Hence, factoring the quadratic expression gives the zeros of the function.

Step 4: Factor \(x^{2} + 9\)

In light of the relationship discovered in Step 3, if asked to factor the quadratic function \(x^{2}+9\), one can recognize that it is in the form of \(x^{2} + c^{2}\) with \(c = 3\). So, similar to the earlier example, the factors will be \((x+3i)\) and \((x-3i)\).

Step 5: Factor a general quadratic function

Following the logic from the previous steps, a general quadratic function in the form \(x^{2} + c^{2}\) where \(c\) is a positive real number can be factored into the expressions \((x+ci)\) and \((x-ci)\). This extends the relationship discovered earlier between factoring a quadratic expression and finding the zeros of the associated function.

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two, and they typically take the form of \(ax^2 + bx + c\). These expressions are central in algebra due to their versatility and appearance across various contexts.

What makes quadratic expressions unique is their graphing into a U-shaped curve known as a parabola. The roots or zeros of these expressions determine where the parabola intersects the x-axis. In complex numbers, quadratic expressions gain extra layers of depth.

Using complex numbers allows us to explore expressions with non-real roots. For instance, the quadratic expression \(x^2 + 1\), when graphed, does not intersect the x-axis in real coordinates. Factoring and understanding these expressions in the complex plane reveals more insights through complex roots.

Factoring

Factoring is the process of breaking down an expression into a product of simpler terms or factors. In the case of quadratic expressions, this means expressing them in the form of \((x - p)(x - q)\), where \(p\) and \(q\) are the roots.

This technique helps us not only in algebra but also in finding solutions to equations that seem unmanageable at first glance. For example, the expression \(x^2 + 9\) can be factored as \((x + 3i)(x - 3i)\). This allows us to solve it efficiently by setting each factor to zero.

The process involves several strategies including "completing the square" or using the quadratic formula, but when working with complex numbers, recognizing patterns like \(a^2 + b^2 = (a + bi)(a - bi)\) becomes essential for factorization.

Zeros of Functions

Zeros of functions, often referred to as roots, are the values that make the function equal to zero. For quadratic functions, these zeros are crucial because they describe the x-coordinates where the graph of the function will touch or cross the x-axis.

For example, the zeros of the function \(f(x) = x^2 + 1\) are \(+i\) and \(-i\). This concept is particularly important when dealing with complex numbers, because it shows how the function behaves in the complex plane—beyond the real number line.

Finding the zeros involves solving the equation \(f(x) = 0\). The relationship between factoring and finding zeros is strong: once you've factored a quadratic expression, the zeros can be directly read from the factors, indicating where \(f(x)\) equals zero.