Problem 18 - Precalculus: Functions and Graphs

Question

Find the zeros of \(f(x),\) and state the multiplicity of each zero. $$f(x)=\left(4 x^{2}-5\right)^{2}$$

Step-by-Step Solution

Answer

Zeros are \(x = \pm \frac{\sqrt{5}}{2}\), each with multiplicity 2.

Step 1: Identify the Expression for Zeros

To find the zeros of the function, set the expression equal to zero: \[ f(x) = (4x^2 - 5)^2 = 0 \] This implies: \[ 4x^2 - 5 = 0 \]

Step 2: Solve for x

Solve the equation \(4x^2 - 5 = 0\) for \(x\).1. Add 5 to both sides to get: \[ 4x^2 = 5 \]2. Divide by 4: \[ x^2 = \frac{5}{4} \]3. Take the square root of both sides: \[ x = \pm \sqrt{\frac{5}{4}} = \pm \frac{\sqrt{5}}{2} \]So the zeros are \(x = \frac{\sqrt{5}}{2}\) and \(x = -\frac{\sqrt{5}}{2}\).

Step 3: Determine Multiplicity of Zeros

The original function was squared, i.e., \((4x^2 - 5)^2\), making each of these zeros have a multiplicity of 2. This means both \(x = \frac{\sqrt{5}}{2}\) and \(x = -\frac{\sqrt{5}}{2}\) are zeros with multiplicity 2.

Key Concepts

Multiplicity of Zeros

The concept of multiplicity gives us insight into whether a zero in a function is a result of a repeated factor. In simpler terms, it tells us how many times a specific solution for a zero appears. If a zero has multiplicity greater than one, it indicates that the line will "touch" the x-axis at that zero but not necessarily "pass through" it.

• If a zero has a multiplicity of 1, it means it crosses the x-axis at that point.
• In our function \(f(x) = (4x^2 - 5)^2\), each zero derived from \(4x^2 - 5 = 0\) had a multiplicity of 2 because the function was squared.
• Thus, both zeros \(x = \frac{\sqrt{5}}{2}\) and \(x = -\frac{\sqrt{5}}{2}\) appear twice in the solutions.

Understanding multiplicity helps in analyzing the behavior of the function around its zeros, aiding in sketching graphs and comprehending function properties.

Quadratic Equation

A quadratic equation is one where the highest degree of the variable is 2, usually written in the form \(ax^2 + bx + c = 0\). Solving these allows us to find the points where the graph of the quadratic function intersects the x-axis. These points are known as the "zeros" or "roots" of the equation.

• The standard methods for solving quadratic equations include factoring, completing the square, and using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• In the exercise, the quadratic \(4x^2 - 5 = 0\) was solved by rearranging terms and applying the square root method.
• This particular approach is useful when the quadratic equation is already presented as a perfect square, allowing us to directly solve for the variable by isolating the square term.

Grasping how to solve quadratic equations is fundamental, as these types of equations appear frequently in various areas of mathematics and applied science.

Solving Equations

Solving equations primarily involves finding the value(s) of the variable(s) that make the equation true. Each type of equation may require a distinct approach:

• For linear equations, we isolate the variable using basic arithmetic operations.
• In quadratic equations, techniques like factoring or the quadratic formula are common.
• Sometimes equations, such as the one in the exercise \((4x^2 - 5)^2 = 0\), involve taking roots after simplifying and isolating terms. This method is efficient when dealing with equations in squared format or higher powers.

It's critical to check solutions by substituting them back into the original equation to ensure correctness. Mastering these techniques aids problem-solving in algebra and other mathematical disciplines, enabling efficient tackling of various problems.