Problem 26 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find all zeros of \(f(x)\) (b) Write the complete factored form of \(f(x)\) $$ f(x)=2 x^{3}+10 x $$

Step-by-Step Solution

Answer

Zeros: \(x = 0, \pm i\sqrt{5}\); Factored form: \(f(x) = 2x(x - i\sqrt{5})(x + i\sqrt{5})\)."}

Step 1: Factor Out the Greatest Common Factor

To begin, look for the greatest common factor in all terms of \(f(x) = 2x^3 + 10x\). Both terms share a factor of \(2x\). Factor this out to simplify the equation: \[ f(x) = 2x(x^2 + 5). \]

Step 2: Solve for All Zeros - First Zero

To find the zeros of the function, set \(2x(x^2 + 5) = 0\). Start with the first factor, \(2x = 0\), and solve for \(x\). Divide both sides by 2: \[ x = 0. \] This is one zero of the function.

Step 3: Solve for All Zeros - Quadratic Factor

Next, focus on the quadratic factor \(x^2 + 5 = 0\). This can be solved by isolating \(x^2\): \[ x^2 = -5. \] Take the square root of both sides: \[ x = \pm \sqrt{-5} = \pm i\sqrt{5}. \] These are the two other zeros of the function, and they are complex.

Step 4: Compile the Complete List of Zeros

The zeros found are \(x = 0\), \(x = i\sqrt{5}\), and \(x = -i\sqrt{5}\). Together, these are all the zeros of the function \(f(x)\).

Step 5: Write the Complete Factored Form

Now express the function with all its factors corresponding to the zeros. The factored form using the zeros is \(f(x) = 2x(x - i\sqrt{5})(x + i\sqrt{5})\). This accounts for all the roots we calculated earlier.

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the building blocks of an algebraic expression. It involves breaking down a complex polynomial into simpler terms that, when multiplied, recreate the original polynomial. This process is vital because it allows us to easily find the zeros, or roots, of the polynomial. When you factor, you can solve for the variable and find where the polynomial equals zero.
In our exercise, we started with the polynomial function:

• Given: \(f(x) = 2x^3 + 10x\)

The first step is finding what the terms share, a process known as factoring out the GCF (Greatest Common Factor). In this case, both terms, \(2x^3\) and \(10x\), share a factor of \(2x\). Once you pull out the GCF, you're left with simpler polynomial terms inside the parentheses:

• Result: \(f(x) = 2x(x^2 + 5)\)

Factoring makes solving equations much more manageable!

Complex Numbers

Complex numbers come into play when we deal with square roots of negative numbers. Normally, you can't take the square root of a negative number and get a real number. Instead, you get an imaginary number. These complex numbers consist of a real part and an imaginary part, and they are useful in solving polynomial equations with no real roots.
In our exercise, after factoring the polynomial, we are left with a quadratic equation inside the parentheses: \(x^2 + 5 = 0\). To find the zeros of this part, you need to solve for \(x\):

• Isolate \(x^2\): \(x^2 = -5\)
• Take the square root: \(x = \pm \sqrt{-5}\)

Since \(\sqrt{-5}\) cannot be a real number, we introduce the imaginary number \(i\), defined as \(i = \sqrt{-1}\). Thus, the solution becomes \(x = \pm i\sqrt{5}\), which are our complex zeros. Complex numbers broaden the scope of solving polynomial equations, allowing us to find all possible roots.

Quadratic Equations

Quadratic equations are polynomials of degree 2, typically in the form \(ax^2 + bx + c = 0\). Solving these equations is critical because they often emerge when factoring polynomials or setting them to zero. They can sometimes provide real solutions, but, as seen in this exercise, they can also yield complex solutions.
In our case, the polynomial includes a quadratic factor: \(x^2 + 5 = 0\). Since the equation lacks a linear term (no \(x\)), solving becomes simpler: just isolate \(x^2\) and then take the square root:

• \(x^2 = -5\)
• \(x = \pm i\sqrt{5}\)

Due to the negative value inside the square root, the solutions are complex. Quadratic equations can, therefore, demonstrate the concept of complex numbers efficiently. Solving them provides critical insight into the nature of polynomial functions.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that divides all terms within the polynomial without a remainder. It's the most significant piece shared between terms and is fundamental when simplifying polynomials through factoring. Discovering the GCF early on helps in breaking down more complicated expressions into easier-to-manage parts.
In our exercise:

• The polynomial was \(2x^3 + 10x\).
• Both terms have the factor \(2x\), the GCF.

Finding the GCF is essential before continuing with solving or factoring further, as it simplifies the polynomial significantly:

• \(f(x) = 2x(x^2 + 5)\)

Factoring out the GCF can transform an otherwise daunting polynomial into a more approachable and straightforward problem, facilitating the process of finding zeros and solving equations.