Problem 56 - College Algebra

Question

For the following exercises, list all possible rational zeros for the functions. \(f(x)=x^{4}+3 x^{3}-4 x+4\)

Step-by-Step Solution

Answer

Possible rational zeros: \(\pm 1, \pm 2, \pm 4\).

Step 1: Identify the Polynomial Function

The polynomial given is \(f(x) = x^4 + 3x^3 - 4x + 4\). This is a fourth-degree polynomial with integer coefficients.

Step 2: State the Rational Root Theorem

According to the Rational Root Theorem, any rational root of the polynomial \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0\) can be written as \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\), and \(q\) is a factor of the leading coefficient \(a_n\).

Step 3: Identify the Constant and Leading Coefficients

For the polynomial \(f(x) = x^4 + 3x^3 - 4x + 4\), the constant term \(a_0\) is 4, and the leading coefficient \(a_4\) is 1.

Step 4: List the Factors of the Constant Term

The factors of the constant term 4 are \(\pm 1, \pm 2,\) and \(\pm 4\).

Step 5: List the Factors of the Leading Coefficient

The factors of the leading coefficient 1 are \(\pm 1\).

Step 6: List All Possible Rational Zeros

Using the Rational Root Theorem, all possible rational zeros are the factors of the constant term divided by the factors of the leading coefficient. Therefore, the possible rational zeros are \(\pm 1, \pm 2, \pm 4\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool for solving polynomial equations. It provides a way to predict possible rational roots of a given polynomial function. Imagine you have a polynomial in the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_0\) with integer coefficients. According to the theorem, any rational solution to the polynomial equation \(f(x) = 0\) can be expressed as \(\frac{p}{q}\). Here, \(p\) is a factor of the constant term \(a_0\), and \(q\) is a factor of the leading coefficient \(a_n\).
To apply this theorem in practice, you'll first identify the constant and leading coefficients of the polynomial. Then, list their factors. For instance, if the constant term is 4, its factors are \(\pm 1, \pm 2, \pm 4\). If the leading coefficient is 1, its factors are \(\pm 1\). After identifying all potential factors, you can create a list of possible rational roots by dividing each factor of the constant term by each factor of the leading coefficient.

Rational Zeros

Rational zeros refer to the roots of a polynomial that can be expressed as a rational number, i.e., a fraction or an integer. These zeros play a significant role in understanding the behavior and graph of polynomial functions. When discussing polynomial roots, it's important to recognize that not all roots have to be rational; however, finding rational ones can simplify further analysis and computation.
Utilizing the Rational Root Theorem, we can list potential rational zeros of polynomials without actually solving the whole polynomial equation. This method narrows down the possibilities and simplifies the trial and error process of finding actual roots. By testing the potential zeros in the polynomial equation, one can determine which are actual zeros. Hence, knowing how to calculate and list these potential zeros is a critical step in solving and graphing polynomial equations efficiently.

Fourth-Degree Polynomial

A fourth-degree polynomial is a polynomial where the highest exponent of the variable is four. It generally takes the form \(ax^4 + bx^3 + cx^2 + dx + e\), with \(a, b, c, d, and e\) being constants and \(a eq 0\). This type of polynomial is also known as a quartic polynomial.
Understanding fourth-degree polynomials is essential because they occur frequently in mathematical modeling and real-world scenarios. They can have at most four roots or solutions, which may be real or complex numbers. The shape of their graphs can vary widely, often featuring unique twists and turns compared to lower-degree polynomials. Analyzing a fourth-degree polynomial involves determining its roots, intercepts, and critical points to understand its behavior fully.
When solving for rational zeros in a fourth-degree polynomial, one often uses methods such as the Rational Root Theorem to simplify initial guesses. Successfully identifying these zeros can make it easier to factor the polynomial and fully solve it, especially when combined with numerical or graphing techniques.