Problem 59 - College Algebra

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 are \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}\).

Step 1: Identify the Leading Coefficient and the Constant Term

The function given is \(f(x)=6 x^{4}-10 x^{2}+13 x+1\). To apply the Rational Root Theorem, first identify the leading coefficient, which is the coefficient of the term with the highest degree (\(6\)), and the constant term, which is \(1\).

Step 2: List the Factors of the Leading Coefficient

List all the factors of the leading coefficient, \(6\). These factors are \(\pm 1, \pm 2, \pm 3, \pm 6\).

Step 3: List the Factors of the Constant Term

List all the factors of the constant term, \(1\). These factors are \(\pm 1\).

Step 4: Apply the Rational Root Theorem

The Rational Root Theorem states that any rational root, \(\frac{p}{q}\), of the polynomial is such that \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. Thus, the potential rational roots are the combination of factors of \(1\) over factors of \(6\).

Step 5: List All Possible Rational Zeros

Using the factors found in the previous steps, list all possible rational zeros of \(f(x)\). These are \(\frac{1}{1}, \frac{-1}{1}, \frac{1}{2}, \frac{-1}{2}, \frac{1}{3}, \frac{-1}{3}, \frac{1}{6}, \frac{-1}{6}\), which simplify to \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}\).

Key Concepts

Leading Coefficient

The leading coefficient is a pivotal element in a polynomial. It affects how the graph of a polynomial behaves, particularly its end behavior. In a polynomial like \(f(x) = 6x^4 - 10x^2 + 13x + 1\), the leading coefficient is the number in front of the term with the highest power of \(x\), which is 6. Understanding the leading coefficient is crucial for applying the Rational Root Theorem, which helps in determining possible rational roots of the polynomial.

When listing possible rational roots, the factors of the leading coefficient play a key role. In this exercise, these factors are \(\pm 1, \pm 2, \pm 3, \pm 6\). Knowing these enables you to comprehend the extent of the possible rational zeros, as each factor could potentially be a divisor in a rational root \(\frac{p}{q}\), where \(q\) is a factor of the leading coefficient. This step sets the stage for further evaluation of potential roots in a polynomial.

Constant Term

The constant term in a polynomial is the term that does not contain any variables, standing alone without any \(x\). In \(f(x) = 6x^4 - 10x^2 + 13x + 1\), the constant term is \(1\). Understanding this term is essential because it is used alongside the leading coefficient to apply the Rational Root Theorem.

For this theorem, we don't just look at the constant term's value; we need to identify its factors. These factors are \(\pm 1\), which will serve as possible values for \(p\) in a rational root \(\frac{p}{q}\), where \(p\) represents the factor of the constant term. With these numbers, one can generate potential rational roots to test whether they are true roots by substituting them into the polynomial.

Factors of a Polynomial

The factors of a polynomial are the building blocks that when multiplied together give the polynomial itself. In the context of the Rational Root Theorem, factors are indispensable because they help suggest potential rational zeros.

The process begins by identifying all factors of the leading coefficient and the constant term. Here, the polynomial \(f(x) = 6x^4 - 10x^2 + 13x + 1\) has leading coefficient factors \(\pm 1, \pm 2, \pm 3, \pm 6\) and constant term factors \(\pm 1\). The combination of these factors according to the theorem implies the possible rational zeros can be expressed as \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}\).

By evaluating these, we can determine if any of these potential zeros will make the polynomial equal zero when substituted in place of \(x\). This helps in simplifying the polynomial or solving it completely by finding its roots.