Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$g(t)=\frac{1}{2} t^{4}-\frac{1}{2}$$

Step-by-Step Solution

Answer

The real zeros of the function are \( t = 1 \) and \( t = -1 \).

Step 1: Set the Function Equal to Zero

Start by setting the function equal to zero to find the values of \( t \) that make the function zero. So the equation to solve is \( \frac{1}{2} t^{4} - \frac{1}{2} = 0 \).

Step 2: Solve the Equation

Next, solve the equation for \( t \). Multiply every term by 2 to get rid of the fraction: \( t^{4} - 1 = 0 \). Add 1 to both sides for \( t^{4} = 1 \). The solutions are \( t = ±\sqrt[4]{1} \) which result in \( t = ±1 \).

Step 3: Confirm Results using a Graphing Utility

Plot the function \( g(t) = \frac{1}{2} t^{4} - \frac{1}{2} \) using a graphing utility. The function should intersect the x-axis at \( t = 1 \) and \( t = -1 \), confirming that these are indeed the zeros of the function.

Key Concepts

Algebraic Solutions

To find the real zeros of a polynomial function algebraically, one must solve for the variable that makes the function equal to zero. In the given exercise, the function is set to zero: \( \frac{1}{2} t^{4} - \frac{1}{2} = 0 \). Here are some key steps and tips to keep in mind:

• Eliminate Fractions: It's easier to handle equations without fractions. Multiply both sides by the denominator, which in this case, clears the fraction and gives us \( t^{4} - 1 = 0 \).
• Rearrange and Simplify: Simple algebraic manipulation like adding or subtracting terms from both sides can help isolate terms. For example, adding 1 to both sides yields \( t^{4} = 1 \).
• Solve for the Variable: You can now solve the equation by taking the fourth root of both sides. Remember, \( t^{4} = 1 \) gives us solutions \( t = \pm 1 \).

By utilizing these strategies, you can confidently solve polynomial equations algebraically.

Graphical Confirmation

Graphical confirmation is a stellar way to verify the solutions you got algebraically. By plotting the polynomial function on a graph, you can visually identify the zeros where the curve crosses the x-axis.Let's break it down:

• Use a Graphing Utility: There are various tools, such as graphing calculators or computer software, to help with plotting functions. For this function \( g(t) = \frac{1}{2} t^{4} - \frac{1}{2} \), inputting it correctly should show the parabola.
• Analyze the Graph: Look for points where the graph intersects the x-axis. These are your zeros. Here, it should show intersections at \( t = 1 \) and \( t = -1 \).

Graphical methods provide a visual check and help in understanding the behavior of the polynomial function across its domain.

Polynomial Functions

Polynomial functions are a fundamental part of algebra. They consist of terms with variables raised to different powers and their coefficients.In the case of our function \( g(t) = \frac{1}{2} t^{4} - \frac{1}{2} \), it is a quartic polynomial. Here's why it matters:

• Degree of the Polynomial: The highest power of the variable, here it's 4, determines the general shape and number of solutions. A quartic polynomial can have up to four real roots. However, in this case, due to the nature of the equation, only \( t = 1 \) and \( t = -1 \) are the real solutions.
• End Behavior: As a polynomial of even degree, the ends of the graph behave similarly, both turning up or down. For \( \frac{1}{2} t^{4} \), both ends rise up as \( t \) becomes very large or very small.

Understanding polynomial functions helps in predicting their graphical behavior and finding their algebraic solutions.