Problem 8 - Precalculus with Calculus Previews

Question

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=3 x^{4}+5 x^{2}+1 $$

Step-by-Step Solution

Answer

The polynomial has no rational zeros.

Step 1: Identify possible rational zeros

To find all rational zeros of the polynomial function \( f(x) = 3x^4 + 5x^2 + 1 \), we apply the Rational Root Theorem. According to the theorem, any possible rational zero is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. Here, the constant term is 1, and the leading coefficient is 3.

Step 2: List the factors of constant and leading coefficient

The factors of the constant term 1 are \( \pm 1 \). The factors of the leading coefficient 3 are \( \pm 1 \) and \( \pm 3 \).

Step 3: Write possible rational zeros

Using all combinations of factors from the Rational Root Theorem, the possible rational zeros are \( \pm 1, \pm \frac{1}{3} \).

Step 4: Test possible zeros using synthetic division or substitution

To determine if any of these possible rational zeros are actual zeros, we substitute them into the polynomial or use synthetic division. Start with \( x = 1 \): \[ f(1) = 3(1)^4 + 5(1)^2 + 1 = 3 + 5 + 1 = 9 \] which is not zero. Similarly, test \( x = -1, \frac{1}{3}, \text{and } -\frac{1}{3} \). For each of these substitutions or synthetic divisions, verify if \( f(x) = 0 \).

Step 5: Conclude that no rational zeros exist

After testing all possible rational zeros, none of them satisfy \( f(x) = 0 \). Therefore, the polynomial function \( f(x) = 3x^4 + 5x^2 + 1 \) has no rational zeros.

Key Concepts

Understanding Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It's a fundamental concept in algebra and forms the basis of many higher-level mathematical topics. The general form of a polynomial function in one variable is:

• \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

where:

• \( a_n, a_{n-1}, \ldots, a_0 \) are constants called coefficients,
• \( n \) is a non-negative integer,
• and \( x \) represents the variable.

In the polynomial \( f(x) = 3x^4 + 5x^2 + 1 \), it includes only terms with non-negative integer exponents, such as \( x^4 \) and \( x^2 \). Each term is a product of a coefficient and a power of \( x \). The highest power of the variable in the expression dictates the degree of the polynomial. Here, the degree is 4, because the term \( 3x^4 \) has the highest exponent. Polynomials are critical because they can be used to model a wide range of real-world phenomena and can undergo operations like addition, subtraction, and multiplication.

Exploring Synthetic Division

Synthetic division is a simplified form of polynomial division, specifically useful when dividing a polynomial by a linear factor of the form \( x - c \). It is generally quicker and requires fewer calculations than long division. Here's a step-by-step outline on how synthetic division works:

• First, write down the coefficients of the dividend (the polynomial you wish to divide). For \( f(x) = 3x^4 + 5x^2 + 1 \), you'd represent missing terms by their coefficients (in this case, "0" for terms like \( x^3 \) and \( x \)). So, the sequence of coefficients becomes 3, 0, 5, 0, 1.
• Next, choose \( c \) from the linear factor \( x - c \), and write it to the left.
• Bring down the leading coefficient as is.
• Multiply \( c \) by this leading coefficient and place the result under the next coefficient. Add these numbers together.
• Repeat the multiplication and addition steps until you reach the last coefficient.
• The final number you get is the remainder, which will be zero if \( x - c \) is a factor of the polynomial.

Using synthetic division simplifies checking potential rational zeros, as it requires fewer steps than substituting values into the polynomial equation each time. This technique is particularly useful when applying the Rational Root Theorem, which predicts the potential rational solutions of a polynomial equation.

The Role of the Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. This is crucial because this value plays a significant role in the behavior of the polynomial's graph, particularly how it stretches or compresses and in determining its end behavior.
In the polynomial \( f(x) = 3x^4 + 5x^2 + 1 \), the leading coefficient is 3. This number is particularly important when applying the Rational Root Theorem. The Rational Root Theorem states that any potential rational zero of the polynomial can be expressed as \( \frac{p}{q} \), where "\( p \)" is a factor of the constant term and "\( q \)" is a factor of the leading coefficient. In this example:

• Factors of the constant term (1) are \( \pm 1 \).
• Factors of the leading coefficient (3) are \( \pm 1 \) and \( \pm 3 \).
• Thus, potential rational roots are \( \pm 1 \) and \( \pm \frac{1}{3} \).

The leading coefficient impacts not just the list of possible rational zeros but also the shape of the graph. For instance, if the leading coefficient is positive, the graph will rise to positive infinity on either side for even-degree polynomials and fall to negative infinity on one side and rise on the other for odd-degree polynomials. Understanding the role of the leading coefficient helps predict and analyze these characteristics effectively.