Problem 5 - Precalculus with Calculus Previews

Question

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=4 x^{4}-7 x^{2}+5 x-1 $$

Step-by-Step Solution

Answer

The rational zero of the polynomial is \(x = \frac{1}{2}\).

Step 1: Identify Potential Rational Zeros Using the Rational Root Theorem

The Rational Root Theorem tells us that any rational zero of the polynomial \(f(x) = 4x^4 - 7x^2 + 5x - 1\) is of the form \(\frac{p}{q}\), where \(p\) divides the constant term and \(q\) divides the leading coefficient. The constant term is -1 with divisors \(\pm 1\), and the leading coefficient is 4 with divisors \(\pm 1, \pm 2, \pm 4\). Thus, the potential rational zeros are \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}\).

Step 2: Evaluate Polynomial at Potential Zeros

To find the real rational zeros, substitute each potential zero into \(f(x)\) and check where the function evaluates to zero.1. \(f(1) = 4(1)^4 - 7(1)^2 + 5(1) - 1 = 1\)2. \(f(-1) = 4(-1)^4 - 7(-1)^2 + 5(-1) - 1 = -7\)3. \(f\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^4 - 7\left(\frac{1}{2}\right)^2 + 5\left(\frac{1}{2}\right) - 1 = 0\)Continue evaluating other potential zeros if needed.

Step 3: Confirm Rational Zero

We found that substituting \(\frac{1}{2}\) into the polynomial results in zero, thus \(\frac{1}{2}\) is a rational zero of \(f(x)\).

Key Concepts

Polynomial Function

A polynomial function is a type of mathematical expression consisting of variables, coefficients, and arithmetic operations like addition, subtraction, and multiplication. These functions are expressed in the form of terms, each with a variable raised to a non-negative integer exponent. The general form of a polynomial function in one variable, say x, is given by: \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \] where:

• \( a_n, a_{n-1}, \ldots, a_0 \) are real number coefficients.
• \( n \) is the degree of the polynomial, which determines the polynomial's highest power.

Each term in a polynomial consists of a coefficient and a variable raised to a power. Polynomials can have any number of terms ranging from a single constant (no variable part) to infinitely many terms. Polynomials play a crucial role in algebra and calculus, forming the building blocks for various mathematical models.

Rational Zeros

Rational zeros, also known as rational roots, are solutions to polynomial equations expressed as fractions of integers. The Rational Root Theorem helps determine these zeros by providing a systematic way to list all possible candidates.The theorem states that if \( \frac{p}{q} \) is a rational zero of a polynomial function \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), then:

• \( p \) is a factor of the constant term \( a_0 \).
• \( q \) is a factor of the leading coefficient \( a_n \).

By using these factors, the potential rational zeros can be found and tested to see which, if any, satisfy the equation when substituted into \( f(x) \). This method simplifies the search for rational solutions by narrowing down the possibilities to a manageable list.

Leading Coefficient

In a polynomial function, the leading coefficient is the coefficient of the term with the highest degree. It provides important information regarding the behavior of the polynomial as the variable approaches large positive or negative values.For the polynomial function \( f(x) = 4x^4 - 7x^2 + 5x - 1 \), the leading term is \( 4x^4 \), and the leading coefficient is 4. Here are several roles played by the leading coefficient in a polynomial:

• The leading coefficient affects the end behavior of the polynomial:
  • If the leading coefficient is positive, the polynomial will tend to infinity as \( x \) approaches infinity.
  • If it is negative, the polynomial will trend towards negative infinity as \( x \) approaches infinity.
• The leading coefficient is used in determining potential rational zeros by dividing into its factors.

This makes the leading coefficient essential in both understanding the graph of the polynomial and finding its rational solutions.

Constant Term

The constant term in a polynomial function is the term without a variable, simply a standalone number. This is the term where the exponent of the variable is zero. In the polynomial function \( f(x) = 4x^4 - 7x^2 + 5x - 1 \), the constant term is -1.The constant term is crucial for the Rational Root Theorem, as it provides the potential values for the numerator \( p \) in the rational equation \( \frac{p}{q} \). Here are important aspects of the constant term:

• It contributes to the y-intercept of the polynomial graph, providing the value of \( f(x) \) when \( x = 0 \).
• When determining rational zeros, the factors of the constant term form the possible values for \( p \) in the rational zero candidates.

In summary, the constant term plays an integral role in both the graphical interpretation and analytical problem-solving approaches in polynomials.