Problem 24 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find all zeros of \(f(x)\) (b) Write the complete factored form of \(f(x)\) $$ f(x)=x^{2}+11 $$

Step-by-Step Solution

Answer

The zeros are \(i\sqrt{11}\) and \(-i\sqrt{11}\); factored form is \((x - i\sqrt{11})(x + i\sqrt{11})\).

Step 1: Identify the Type of Function

The given function is a quadratic function of the form \( f(x) = x^2 + 11 \). This means we are looking for values of \( x \) where the function equals zero.

Step 2: Set the Function to Zero

Set the equation \( f(x) = x^2 + 11 \) to zero to find its zeros: \( x^2 + 11 = 0 \).

Step 3: Solve for \( x \)

Rearrange the equation \( x^2 + 11 = 0 \) to find \( x^2 = -11 \). Take the square root of both sides of the equation to solve for \( x \), yielding \( x = \pm \sqrt{-11} \). Since \( \sqrt{-11} \) is an imaginary number, we express this as \( x = \pm i\sqrt{11} \). Thus, the zeros are \( x = i\sqrt{11} \) and \( x = -i\sqrt{11} \).

Step 4: Write the Complete Factored Form

Using the zeros found, the factored form of the quadratic function can be expressed as \( f(x) = (x - i\sqrt{11})(x + i\sqrt{11}) \). This reflects the fact that the function has imaginary roots.

Key Concepts

Imaginary Numbers

When dealing with quadratic functions, sometimes you'll encounter solutions that aren't real numbers. In these cases, we use imaginary numbers. An imaginary number is defined as a multiple of the imaginary unit, represented by \( i \). The imaginary unit satisfies the equation \( i^2 = -1 \).

So, when you have a negative number under a square root, such as \( \sqrt{-11} \), it becomes an imaginary number. By expressing it as \( \sqrt{-11} = i\sqrt{11} \), we convert it into an expression involving \( i \).

This enables us to handle complex solutions in a systematic way. In our problem, the roots \( x = i\sqrt{11} \) and \( x = -i\sqrt{11} \) are complex because they involve imaginary numbers.

Factoring Polynomials

Factoring polynomials is crucial in simplifying algebraic expressions and finding solutions, especially when working with quadratics. A polynomial is factored when it is expressed as a product of its roots or simpler polynomials.

In our exercise, the quadratic function \( f(x) = x^2 + 11 \) can be factored based on its roots found earlier. Understanding that these roots are \( x = i\sqrt{11} \) and \( x = -i\sqrt{11} \), we can write the function as a product:

• \( f(x) = (x - i\sqrt{11})(x + i\sqrt{11}) \)

This expression is the complete factored form of the polynomial. It's useful because it clearly presents the roots of the equation.

Finding Zeros

Finding the zeros of a quadratic function is like answering the question "For which values of \( x \) does the function equal zero?". These values correspond to the roots of the equation.

For the function \( f(x) = x^2 + 11 \), we set it equal to zero:

• \( x^2 + 11 = 0 \)

Rearranging gives us \( x^2 = -11 \), and finding the square root leads to imaginary solutions. These zeros, \( x = i\sqrt{11} \) and \( x = -i\sqrt{11} \), highlight that sometimes the solutions lie in the complex plane rather than the real one.

Identifying the zeros helps in building the factorized form and understanding the nature of the roots, whether real or imaginary.