Problem 10 - Precalculus

Question

List the possibilities for rational roots. $$\frac{1}{2} x^{4}-5 x^{3}+\frac{4}{3} x^{2}+8 x-\frac{1}{3}=0$$

Step-by-Step Solution

Answer

The possible rational roots are ±1, ±1/3, ±2, ±2/3, ±1/2, ±1/6, ±3/2, ±1/4.

Step 1: Identify the Coefficients

For the polynomial \( \frac{1}{2} x^4 - 5x^3 + \frac{4}{3} x^2 + 8x - \frac{1}{3} = 0 \), identify the coefficients of the terms: \( a = \frac{1}{2} \), \( b = -5 \), \( c = \frac{4}{3} \), \( d = 8 \), and \( e = -\frac{1}{3} \).

Step 2: Apply Rational Root Theorem

According to the Rational Root Theorem, any rational solution, \( \frac{p}{q} \), to the polynomial must have \( p \) as a factor of the constant term (-\frac{1}{3}) and \( q \) as a factor of the leading coefficient (\frac{1}{2}).

Step 3: List Factors of the Constant Term

The factors of the constant term \(-\frac{1}{3}\) are \( \pm 1 \) and \( \pm \frac{1}{3} \).

Step 4: List Factors of the Leading Coefficient

The factors of the leading coefficient \( \frac{1}{2} \) are \( \pm 1 \), \( \pm \frac{1}{2} \).

Step 5: Combine and Simplify Factors

Create rational roots by making fractions \( \frac{p}{q} \) from the combinations of the factors of the constant term \(-\frac{1}{3}\) and the factors of the leading coefficient \( \frac{1}{2} \). The possible rational roots are: \( \pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm \frac{1}{2}, \pm \frac{1}{6}, \pm \frac{3}{2}, \pm \frac{1}{4} \).

Key Concepts

Polynomial Coefficients

In any polynomial, the coefficients are those numbers placed in front of the variables. These coefficients determine many of the properties of the polynomial. For the polynomial \(\frac{1}{2} x^4 - 5x^3 + \frac{4}{3} x^2 + 8x - \frac{1}{3} = 0\), these numbers are found preceding each term and essentially define the polynomial's behavior.
Detecting each coefficient is crucial for further analysis:

• The first coefficient, \( a = \frac{1}{2} \), applies to \( x^4 \).
• Following the pattern, \( b = -5 \), is paired with \( x^3 \).
• Next is \( c = \frac{4}{3} \) with \( x^2 \).
• Then, \( d = 8 \) for \( x \).
• Finally, \( e = -\frac{1}{3} \) represents the constant term.

Each coefficient contributes to forming the solution of the polynomial equation by impacting things like the degree and the potential transformations of the graph.

Factors of Constant Term

The constant term is the part of the polynomial that doesn't wield a variable, such as \(-\frac{1}{3}\) in this case. When applying the Rational Root Theorem, we focus on the constant term to predict the polynomial's possible rational roots.
To achieve this, we find the factors of the constant term. Factors are simply numbers that divide evenly into another number without leaving a remainder. For \(-\frac{1}{3}\), the factors are:

• \(\pm 1\)
• \(\pm \frac{1}{3}\)

These potential values of \(p\) are utilized to construct rational solutions to the polynomial through the Rational Root Theorem, playing an integral part in calculating possible root values.

Factors of Leading Coefficient

The leading coefficient in this polynomial is the number attached to the highest degree term, which is \( \frac{1}{2} \) for \( x^4 \). Identifying its factors is significant in using the Rational Root Theorem.
Factors of the leading coefficient are essential as they act as potential denominators \(q\) for the rational root equation \(\frac{p}{q}\). Hence, the factors of \(\frac{1}{2}\) are:

• \(\pm 1\)
• \(\pm \frac{1}{2}\)

Using these factors, one can form different combinations with the factors of the constant term. This step completes the preparation for testing possible rational root values, which can reflect solutions of the polynomial if verified.

Rational Roots

Rational roots are the solutions to a polynomial equation that can be expressed as the ratio of two integers, \(\frac{p}{q}\). The Rational Root Theorem is a helpful tool to predict the potential rational roots for those wanting to solve polynomials.
According to the theorem, the numerator \(p\) must be a factor of the constant term, and the denominator \(q\) must be a factor of the leading coefficient. In our exercise, these combinations form the possible roots as follows:

• \(\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm \frac{1}{2}, \pm \frac{1}{6}, \pm \frac{3}{2}, \pm \frac{1}{4}\)

Testing these values in the polynomial will reveal which, if any, are true roots. Only through computation and substitution can one confirm which possible value is indeed a solution, contributing to the determination of real and practical applications of polynomial functions.