Problem 6 - Precalculus

Question

List the possibilities for rational roots. $$x^{4}-x^{3}+10 x^{2}-24=0$$

Step-by-Step Solution

Answer

Possible rational roots are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\).

Step 1: Identify the Constant and Leading Coefficient

In the polynomial equation, identify the constant term and the leading coefficient. The constant term is the term without any variable, which is \(-24\) in this case, and the leading coefficient is the coefficient of the term with the highest degree, which is \(1\) for \(x^4\).

Step 2: List Factors of the Constant Term

Find all integer factors of the constant term, \(-24\). The factors are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\).

Step 3: List Factors of the Leading Coefficient

Find all integer factors of the leading coefficient, \(1\). Since the leading coefficient is \(1\), its factors are \(\pm 1\).

Step 4: Formulate Possible Rational Roots

According to the Rational Root Theorem, possible rational roots are given by \(\frac{p}{q}\), where \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient. Since the only factor of \(1\) is \(\pm 1\), the possible rational roots are simply the factors of the constant term, \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\).

Key Concepts

Polynomial Equation

A polynomial equation is a mathematical expression that involves variables and coefficients, arranged using the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, the polynomial equation given is \(x^4 - x^3 + 10x^2 - 24 = 0\). Here, each term is composed of a variable \(x\) raised to a power and multiplied by a coefficient. The highest power of the variable is called the degree of the polynomial, which in this case is 4. This indicates that it is a fourth-degree polynomial equation.
The goal when dealing with polynomial equations is often to find the values of \(x\) that make the equation true. These values are known as the roots of the polynomial. The Rational Root Theorem is a handy tool used to find potential rational roots of a polynomial equation. It helps by providing a set of possible values that can be tested to find actual roots.

Constant Term

The constant term in a polynomial equation is the term that contains no variable; it's just a number. In the polynomial equation \(x^4 - x^3 + 10x^2 - 24 = 0\), the constant term is \(-24\).
This term plays a crucial role when using the Rational Root Theorem. The theorem states that any possible rational root, when expressed as a fraction \(\frac{p}{q}\), has a numerator \(p\) that is a factor of the constant term. This makes finding factors of the constant term an essential step in identifying potential rational roots.
Finding these integer factors is a straightforward process whereby we list all possible numbers that divide the constant term (\(-24\) in our example) without leaving a remainder. These factors,

• \(\pm 1\)
• \(\pm 2\)
• \(\pm 3\)
• \(\pm 4\)
• \(\pm 6\)
• \(\pm 8\)
• \(\pm 12\)
• \(\pm 24\)

give us a set of values to work with when finding rational roots.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In our polynomial equation \(x^4 - x^3 + 10x^2 - 24 = 0\), the term with the highest degree is \(x^4\), and its coefficient is 1.
The leading coefficient is important for determining the possible rational roots using the Rational Root Theorem. In the expression \(\frac{p}{q}\), \(q\) should be a factor of the leading coefficient. Since our leading coefficient is 1, the only factors are \(\pm 1\). This simplifies the combinations of potential rational roots because there are no other integers that can divide 1 evenly.
With a leading coefficient of 1, finding the rational roots is slightly easier since it limits the possible values of \(q\) drastically, merging the set of numerators to the integer factors of the constant term.

Integer Factors

Integer factors are whole numbers that can be multiplied together to produce another number. Understanding integer factors is crucial when working with the Rational Root Theorem.
In the context of our polynomial equation \(x^4 - x^3 + 10x^2 - 24 = 0\), we need to find the integer factors of both the constant term and the leading coefficient. The integer factors of the constant term \(-24\) are those numbers that can be divided into \(-24\) without leaving a remainder. These include:

• \(+ 1, -1\)
• \(+ 2, -2\)
• \(+ 3, -3\)
• \(+ 4, -4\)
• \(+ 6, -6\)
• \(+ 8, -8\)
• \(+ 12, -12\)
• \(+ 24, -24\)

The factors of the leading coefficient (which is 1) are simply \(\pm 1\).
Calculating these factors allows us to apply the Rational Root Theorem, providing a clear list of potential roots we can test to find the actual rational solutions.