Problem 30 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. Use a calculator to solve if necessary. By checking only the equation and the coefficients, determine the smallest and largest possible rational roots of the equation \(2 x^{4}+x^{2}+22 x+26=0\).

Step-by-Step Solution

Answer

The smallest possible rational root is \(-26\) and the largest is \(26\).

Step 1: Identify the Rational Root Theorem

The Rational Root Theorem states that any rational solution, expressed as a fraction \( \frac{p}{q} \), of a polynomial equation \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 = 0 \) must have \( p \) as a factor of the constant term \( a_0 = 26 \) and \( q \) as a factor of the leading coefficient \( a_n = 2 \).

Step 2: List the Factors

First, list all the factors of the constant term (26): \( \pm 1, \pm 2, \pm 13, \pm 26 \). Next, list the factors of the leading coefficient (2): \( \pm 1, \pm 2 \).

Step 3: Determine Possible Rational Roots

The possible rational roots are formed by \( \frac{p}{q} \), where \( p \) is a factor of 26 and \( q \) is a factor of 2. List them: \( \pm 1, \pm 2, \pm 13, \pm 26, \pm \frac{1}{2}, \pm \frac{13}{2} \).

Step 4: Order the Rational Roots

To find the smallest and largest rational roots, order them: \(-26, -13, -2, -1, -\frac{13}{2}, -\frac{1}{2}, \frac{1}{2}, 1, 2, 13, 26\).

Step 5: Identify the Smallest and Largest Rational Roots

From the ordered list, the smallest possible rational root is \(-26\) and the largest possible rational root is \(26\).

Key Concepts

Polynomial Equations

Polynomial equations, like the one given in the exercise, are mathematical expressions that equate a polynomial to zero. A typical polynomial has terms made up of variables raised to whole-number powers, multiplied by coefficients. For example, the polynomial equation in the exercise is a fourth-degree polynomial since the highest power of the variable \(x\) is 4: \(2x^4 + x^2 + 22x + 26 = 0\).
These equations may include different terms with various degrees. Solving such equations means finding the roots or values of \(x\) that make the equation equal to zero. One useful method for finding rational solutions is the Rational Root Theorem.
Understanding polynomial equations is crucial because they appear frequently in various fields, such as physics and engineering, representing phenomena like the trajectory of objects. When dealing with these equations, identifying the correct polynomial degree and using appropriate solving techniques is vital.

Factors of Constants

Factors are numbers that, when multiplied together, produce another number. In polynomial equations, the constant term is the fixed number without any variable attached—in this case, 26. The Rational Root Theorem suggests that any rational root of the equation must have a numerator that is a factor of this constant term.
To find these factors, break down the constant into all its divisors. For 26, these factors include \(\pm 1, \pm 2, \pm 13,\) and \(\pm 26\). These numbers are significant because they can potentially help identify candidates for rational solutions to the equation.
Understanding the role of constants and their factors is a key part of using the Rational Root Theorem effectively. This knowledge aids in narrowing down the possible rational solutions and simplifies the process of solving polynomial equations.

Leading Coefficients

In polynomial equations, the leading coefficient is the number attached to the term with the highest power. It essentially "leads" the polynomial and heavily influences the overall shape of the graph of the equation. For our polynomial \(2x^4 + x^2 + 22x + 26 = 0\), the leading coefficient is 2 because it is the coefficient of the highest power, \(x^4\).
The Rational Root Theorem indicates that the denominator of any possible rational root must be a factor of this leading coefficient. Thus, we consider the factors of 2, which include \(\pm 1\) and \(\pm 2\).
Identifying the leading coefficient and understanding its factors are crucial steps in applying the Rational Root Theorem. It helps to limit and organize the possible rational solutions of the equation, making the process of solving polynomial equations more manageable and systematic.