Problem 59 - College Algebra with Corequisite Support

Question

For the following exercises, list all possible rational zeros for the functions. $$ f(x)=6 x^{4}-10 x^{2}+13 x+1 $$

Step-by-Step Solution

Answer

Possible rational zeros: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6} \).

Step 1: Identify the Leading and Constant Coefficients

For the polynomial \( f(x) = 6x^4 - 10x^2 + 13x + 1 \), the leading coefficient is \( 6 \) and the constant term is \( 1 \). These coefficients will help us determine possible rational zeros.

Step 2: Find Factors of the Constant Term

List all positive and negative factors of the constant term, which is \( 1 \). The factors are \( \pm 1 \).

Step 3: Find Factors of the Leading Coefficient

List all positive and negative factors of the leading coefficient, \( 6 \). The factors are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Step 4: Use the Rational Root Theorem

According to the Rational Root Theorem, possible rational zeros are given by \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. Thus, possible rational zeros are \( \pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6} \).

Step 5: List All Possible Rational Zeros

The complete list of possible rational zeros is: \[ \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6} \].

Key Concepts

Polynomial Functions

A polynomial function is a type of mathematical expression that forms the backbone of many algebraic operations. It is composed of variables and coefficients linked together by only addition, subtraction, and multiplication. In our given example, the polynomial function is expressed as \[ f(x) = 6x^4 - 10x^2 + 13x + 1 \].
Each term in this polynomial consists of a coefficient and a variable raised to a non-negative integer power.

• The term \(6x^4\) represents a variable \(x\) raised to the power of 4, with a coefficient of 6.
• \(-10x^2\) involves the term \(x^2\) multiplied by -10.
• \(13x\) shows that \(x\) alone is multiplied by 13.
• The constant term is \(1\), standing by itself.

Polynomial functions can have various degrees, which is determined by the highest power of the variable present. For the given function, the degree is 4 because the highest exponent is 4.

Rational Root Theorem

The Rational Root Theorem is a powerful tool in algebra that assists in identifying possible rational roots, or zeros, of a polynomial equation. This theorem states that any possible rational solution to a polynomial equation, \[ a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 = 0 \] can be written as a fraction \( \frac{p}{q} \). Here, \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.
In our step-by-step problem, we apply it to the polynomial \[ f(x) = 6x^4 - 10x^2 + 13x + 1 \],
identifying the rational zeros by dividing the factors of the constant term, \(1\), by the factors of the leading coefficient, \(6\), giving us possible zeros such as \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6} \).
The theorem provides a systematic method to list potential rational roots, simplifying the root-finding process for polynomials.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the behavior of the polynomial, particularly its growth and end behavior. In the polynomial \[ f(x) = 6x^4 - 10x^2 + 13x + 1 \],
the leading coefficient is \(6\), associated with the term \(6x^4\), which is the highest-degree term present.

• The leading coefficient affects the polynomial’s growth rate, influencing how steeply it inclines or declines.
• When applying the Rational Root Theorem, the leading coefficient's factors help determine possible rational zeros.

Without understanding the leading coefficient, we can't fully explore the characteristics and potential solutions of polynomial functions.

Constant Term

The constant term in a polynomial is the term without any variable attached to it. It can be thought of as the y-intercept in a graph of the function, representing the value of the polynomial when the variable is zero. In our example function \[ f(x) = 6x^4 - 10x^2 + 13x + 1 \],
the constant term is \(1\). Here’s why it’s important:

• In the Rational Root Theorem, the constant term’s factors help to derive possible values for \(p\) in \(\frac{p}{q}\), identifying potential rational roots.
• Its sign can help predict the overall behavior of the polynomial, impacting whether the function has even or odd number of real roots.

The constant term is as significant as any other term, providing crucial information in analyzing and solving polynomial equations.