Problem 40 - A Graphical Approach to College Algebra

Question

Without graphing, answer true or false to each statement. Then, support your answer by graphing. The function \(f(x)=3 x^{4}+5\) has no real zeros.

Step-by-Step Solution

Answer

True, the function has no real zeros.

Step 1: Understand the Problem

We need to determine if the function \( f(x) = 3x^4 + 5 \) has any real zeros. A real zero of a function is a value of \( x \) for which the function equals zero, i.e., \( f(x) = 0 \).

Step 2: Set the Function Equal to Zero

To find the zeros, set the function equal to zero: \[ 3x^4 + 5 = 0 \] Solve for \( x \).

Step 3: Isolate the Polynomial Term

Subtract 5 from both sides to isolate the polynomial term:\[ 3x^4 = -5 \]

Step 4: Analyze the Equation

Divide both sides by 3 to solve for \( x^4 \):\[ x^4 = -\frac{5}{3} \] Real numbers raised to an even power result in a non-negative value. Therefore, \( x^4 \) cannot be negative.

Step 5: Draw a Conclusion

Since \( x^4 = -\frac{5}{3} \) has no real solutions (as \( x^4 \) cannot be negative), the function \( f(x) = 3x^4 + 5 \) has no real zeros.Hence, it is true that the function has no real zeros.

Key Concepts

Understanding Polynomial Functions

Polynomial functions are mathematical expressions that involve sums of powers of a variable. For example, the function \( f(x) = 3x^4 + 5 \) is a polynomial. The highest power of the variable indicates the degree of the polynomial. Here, the degree is 4. We call this a fourth-degree polynomial.
A key feature of polynomials is that they are defined for all real numbers. This means that for any real number you substitute for \( x \), you'll get a real number as the result. Polynomials can have real zeros, imaginary zeros, or no zeros at all depending on their structure. Zeros of a polynomial are the values of \( x \) that make the entire polynomial equal to zero.

Exploring Even Powers in Polynomial Functions

The concept of an even power is simple yet crucial in determining the characteristics of polynomial functions. An even power occurs when the variable is raised to an even number, such as 2, 4, 6, etc. In the function \( f(x) = 3x^4 + 5 \), the term \( x^4 \) represents an even power.

• Even powers of real numbers are always non-negative; they result in positive numbers unless the base is zero.
• This is due to squaring or raising any number to an even power, which produces a positive result.

Seeing this term \( x^4 \), we understand that the output of \( f(x) \) involving this power will always be positive when added to 5. Therefore, any equation set to zero cannot be solved using real numbers if the other side of the equation is negative.

Solving Polynomial Equations

Solving polynomial equations involves finding the value of \( x \) that makes the polynomial equal zero, known as the roots or zeros of the polynomial. Let's see how we apply this to our example.
In our equation \( 3x^4 + 5 = 0 \), we start by isolating the polynomial term. Follow these simple steps:

• Subtract 5 from both sides, leading to \( 3x^4 = -5 \).
• Divide both sides by 3 to simplify: \( x^4 = -\frac{5}{3} \).

At this point, we recognize that no real number can make \( x^4 = -\frac{5}{3} \) true, as \( x^4 \) is an even power and must be non-negative. Therefore, in this case, the equation has no real solutions.

Analyzing Functions to Determine Zeros

Analyzing a polynomial function helps us understand its behavior and possible zeros. We look at the basic form and degree to interpret its graph and solve related equations.
The function \( f(x) = 3x^4 + 5 \) is always positive because the smallest value \( 3x^4 \) can reach is zero, adding this to 5 gives a positive number. Here's how to conclude from analysis:

• Recognize that the even degree signs the function to open upwards, similar to a U-shape when you graph it.
• Since the base polynomial \( x^4 \) is non-negative, the lowest point of \( f(x) \) is greater than zero.
• Therefore, \( f(x) \) will never intersect the x-axis; it confirms no real zeros exist.

By analyzing the degree and leading coefficient of the polynomial, we predict graphical behavior and discuss the possibility of zeros.