Problem 23 - Algebra for College Students

Question

For Problems \(21-26\), verify that the equations do not have any rational number solutions. $$ 3 x^{4}-4 x^{3}-10 x^{2}+3 x-4=0 $$

Step-by-Step Solution

Answer

The equation has no rational number solutions.

Step 1: Understand Rational Root Theorem

The Rational Root Theorem states that any possible rational root, in the form of \( \frac{p}{q} \), of the polynomial equation \( a_nx^n + a_{n-1}x^{n-1} + ... + a_0 = 0 \) is a fraction where \( p \) is a factor of the constant term \( a_0 \) and \( q \) is a factor of the leading coefficient \( a_n \). For this polynomial \( 3x^4 - 4x^3 - 10x^2 + 3x - 4 = 0 \), the constant term is -4 and the leading coefficient is 3.

Step 2: Identify Possible Rational Roots

List all the factors of the constant term \(-4\): \( \pm 1, \pm 2, \pm 4 \). List all the factors of the leading coefficient \(3\): \( \pm 1, \pm 3 \). The possible rational roots are all combinations of these factors in \( \frac{p}{q} \) form: \( \pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3} \).

Step 3: Evaluate Polynomial at Possible Roots

Substitute each possible rational root into the polynomial equation \( 3x^4 - 4x^3 - 10x^2 + 3x - 4 \) and calculate the value. If the result is not zero for any \/ all the possible values, then no rational roots exist.

Step 4: Verify by Substitution

For each candidate (e.g., \( x = 1, x = -1, x = 2, \) etc.), substitute into the polynomial:- For \( x = 1:\) \( 3(1)^4 - 4(1)^3 - 10(1)^2 + 3(1) - 4 = -12 \) (not zero)- For \( x = -1:\) \( 3(-1)^4 - 4(-1)^3 - 10(-1)^2 + 3(-1) - 4 = -18 \) (not zero)- Continue this for all candidates. If none equates to zero, no rational roots exist.

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions involving variables raised to various powers, combined with coefficients. These equations take the form of a sum of terms, for example, a general polynomial is expressed as:

• \( a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_0 = 0 \)

Here, \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( n \) represents the highest degree of the equation.
The degree of the polynomial indicates the highest power of the variable involved. Higher-degree polynomials can have more complex solutions, involving both real and complex numbers.
Understanding polynomial equations is essential as they form the foundation of many algebraic concepts. They are used to describe a wide range of phenomena and appear in various fields such as physics, engineering, and economics.

Rational Number Solutions

In the context of polynomial equations, a rational number solution is a solution to the equation that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q eq 0 \).
The Rational Root Theorem aids in identifying potential rational solutions to polynomial equations. If a polynomial has a rational root, it is a fraction where the numerator is a factor of the constant term, and the denominator is a factor of the leading coefficient.
Finding rational solutions is often one of the first steps to solving polynomial equations, helping to simplify the problem. If no rational number solutions exist, more complex techniques, such as factoring, graphing, or numerical methods, might be necessary to determine the roots.

Factors of Coefficients

Understanding the factors of the coefficients, particularly the constant term and the leading coefficient, is a crucial part of applying the Rational Root Theorem.

• Factors of a number are the integers that can be multiplied together to obtain that number.
• For a constant term like \(-4\), its factors are \( \pm 1, \pm 2, \pm 4 \).
• For a leading coefficient like \(3\), its factors are \( \pm 1, \pm 3 \).

These factors help determine the possible rational roots by combining them into fractions. Knowing the factors allows us to list all possible \( \frac{p}{q} \) combinations, which are then checked against the polynomial equation.
Thus, the factors serve as key components in assessing the possible rational solutions and narrowing down the search for potential roots.

Polynomial Substitution

Polynomial substitution is a method used for verifying potential solutions to a polynomial equation. Once possible rational roots are identified using the Rational Root Theorem, the next step is to verify them.
To substitute means to replace the variable in the equation with a given number. This involves plugging each of the possible rational roots into the polynomial to evaluate whether they satisfy the equation. The procedure is straightforward:

• Substitute a candidate root into the entire polynomial equation.
• Calculate the outcome.
• If the result is zero, the candidate is a root; otherwise, it is not.

This step is crucial because merely listing possible roots isn't enough. Substitution confirms which, if any, are actual solutions. In exercises like verifying polynomial solutions, substitution is indispensable, as it finalizes whether any rational solutions exist.