Problem 56 - College Algebra with Corequisite Support

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 are \(\pm 1, \pm 2, \pm 4\).

Step 1: Identify the Leading Coefficient and Constant Term

Examine the polynomial function given, which is \(f(x) = x^4 + 3x^3 - 4x + 4\). Here, the leading coefficient is the coefficient of the highest power of \(x\), which is 1 for \(x^4\). The constant term is the standalone number at the end, which is 4.

Step 2: Use the Rational Root Theorem

The Rational Root Theorem states that any rational solution, expressed as \(\frac{p}{q}\), to a polynomial function is such that \(p\) (the numerator) divides the constant term and \(q\) (the denominator) divides the leading coefficient.

Step 3: List Factors of the Constant Term

Find all positive and negative factors of the constant term, 4. They are \(\pm 1, \pm 2, \pm 4\).

Step 4: List Factors of the Leading Coefficient

Find all positive and negative factors of the leading coefficient, 1. They are \(\pm 1\).

Step 5: Write Possible Rational Zeros

Apply the Rational Root Theorem. The possible rational zeros are \(\frac{p}{q}\) where \(p\) are factors of the constant term (\(\pm 1, \pm 2, \pm 4\)) and \(q\) are factors of the leading coefficient (\(\pm 1\)). By dividing each factor of 4 by each factor of 1, the possible rational zeros are \(+1, -1, +2, -2, +4, -4\).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The structure of a polynomial can vary from the simple linear form to higher degrees, such as quadratic, cubic, quartic, and so on.
Polynomials are both simple and powerful, making them a fundamental building block in algebra.

• The degree of a polynomial is determined by the highest power of the variable, known as the leading term.
• The general form of a polynomial is expressed as: \[a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\]where each "a" represents a coefficient.
• Polynomials can be added, subtracted, multiplied, and, to some extent, divided using algebraic techniques.

Understanding polynomial functions is essential because they offer a means to represent and solve real-world scenarios by modeling complex relationships in a simplified manner.

Constant Term

The constant term in a polynomial is the term that does not contain any variables. In other words, it stands alone as a fixed number.
The constant term is an integral part of the polynomial, and it often plays a crucial role in finding solutions like rational zeros using the Rational Root Theorem.

• In the polynomial, \(f(x) = x^4 + 3x^3 - 4x + 4\), the constant term is 4.
• It provides a horizontal shift to the graph of the polynomial.
• The constant term can often give insights into the y-intercept of the polynomial function's graph, where the function crosses the y-axis.

In the context of rational zeros, the divisors of the constant term become potential numerators for any rational zero solutions.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a significant role in determining the overall shape of the graph of the polynomial and impacts the properties of the polynomial, such as end behavior.

• In the polynomial \(f(x) = x^4 + 3x^3 - 4x + 4\), the leading coefficient is 1 (the coefficient of the term \(x^4\)).
• If the leading coefficient is positive, the graph will typically rise on the right side, while if it's negative, the graph will fall on the right side.
• For the Rational Root Theorem, the leading coefficient’s factors are used as possible denominators for rational zeros.

The leading coefficient ensures simplicity in determining possible rational zeros; for example, when it is 1, every potential rational zero is just a factor of the constant term.

Rational Zeros

Rational zeros are solutions to a polynomial equation that can be expressed as a fraction where both the numerator and denominator are integers. Finding rational zeros involves applying the Rational Root Theorem.
This theorem helps identify potential zeros without resorting to exhaustive trial-and-error methods.

• The Rational Root Theorem states that any rational zero of a polynomial, \( \frac{p}{q} \), results from \( p \) dividing the constant term and \( q \) dividing the leading coefficient.
• For \(f(x) = x^4 + 3x^3 - 4x + 4\), the possible rational zeros are determined by dividing the factors of 4 (constant term) by the factors of 1 (leading coefficient), yielding \(+1, -1, +2, -2, +4, -4\).
• This list of possible zeros allows for efficient testing to establish which ones are actual roots of the polynomial.

By evaluating these possibilities, one can determine the actual rational zeros, thus fully solving the polynomial function for its rational solutions.