Problem 16 - Precalculus

Question

Show that each equation has no rational roots. $$4 x^{5}-x^{4}-x^{3}-x^{2}+x-8=0$$

Step-by-Step Solution

Answer

The polynomial has no rational roots.

Step 1: Recall the Rational Root Theorem

According to the Rational Root Theorem, any rational root of a polynomial equation, represented as \( \frac{p}{q} \), must satisfy the condition where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. For the given polynomial \( 4x^5 - x^4 - x^3 - x^2 + x - 8 = 0 \), the constant term is \(-8\) and the leading coefficient is \(4\). Hence, the possible values of \( p \) are ±1, ±2, ±4, ±8 and the possible values of \( q \) are ±1, ±2, ±4.

Step 2: List all Possible Rational Roots

By applying the Rational Root Theorem, the possible rational roots are found by forming fractions \( \frac{p}{q} \) using the factors from Step 1. These fractions are: ±1, ±2, ±4, ±8, ±\(\frac{1}{2}\), ±\(\frac{1}{4}\), ±\( \frac{2}{4} \) (simplifies to ±\( \frac{1}{2} \)), ±\( \frac{4}{1} \) (simplifies to ±4), and similarly for other combinations. The complete list includes: ±1, ±2, ±4, ±8, ±\(\frac{1}{2}\), ±\(\frac{1}{4}\).

Step 3: Test Each Possible Rational Root

Substitute each possible root into the polynomial to check if it results in zero. Substitute ±1, ±2, ±4, ±8, ±\(\frac{1}{2}\), and ±\(\frac{1}{4}\) into the polynomial. Use substitution to evaluate whether \(4(\text{root})^5 - (\text{root})^4 - (\text{root})^3 - (\text{root})^2 + (\text{root}) - 8 = 0\). Each substitution should not yield zero, indicating that none of these are roots.

Step 4: Conclude the Lack of Rational Roots

From the testing in Step 3, if none of the substitutions yield zero, we conclude that the polynomial has no rational roots. Since we exhaustively tested all possible rational roots derived from the Rational Root Theorem, the lack of zero results confirms there are no rational roots for the polynomial.

Key Concepts

Polynomial Equations

A polynomial equation is a mathematical expression that involves a polynomial, which can contain multiple terms made up of variables raised to whole number exponents and multiplied by coefficients. An example of a polynomial equation is given by:

• \( ax^n + bx^{n-1} + ... + k = 0 \)

Here, \( n \) represents the degree of the polynomial, which is determined by the highest power of the variable \( x \). Coefficients \( a, b, ..., k \) are constant values associated with each term. Understanding the structure of these equations is crucial for applying various theorems, such as the Rational Root Theorem, which helps in determining possible rational solutions to polynomial equations. In our exercise, the equation \( 4x^5 - x^4 - x^3 - x^2 + x - 8 = 0 \) is a 5th degree polynomial. This implies there could be up to 5 roots, which may be rational or irrational, real or complex.

Factors of Constant Term

When applying the Rational Root Theorem, identifying the factors of the constant term, which is the term that does not include any variable, is crucial. In our polynomial, \( 4x^5 - x^4 - x^3 - x^2 + x - 8 \), the constant term is \(-8\).

• The factors of \(-8\) are: \( ±1, ±2, ±4, ±8 \).

These factors are potential numerators \( p \), in the rational root formula \( \frac{p}{q} \). Understanding the role of these factors helps in efficiently screening for possible rational solutions. By trying each factor as a potential numerator, we can determine which, if any, makes the polynomial equation equal zero, thereby confirming a rational root.

Factors of Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest power of the variable. In the polynomial \( 4x^5 - x^4 - x^3 - x^2 + x - 8 \), the leading term is \( 4x^5 \), making the leading coefficient \( 4 \).

• The factors of \( 4 \) are: \( ±1, ±2, ±4 \).

These factors serve as potential denominators \( q \) when utilizing the Rational Root Theorem formula \( \frac{p}{q} \). By dividing the factors of the constant term by the factors of the leading coefficient, we get a list of potential candidates for rational roots. Properly understanding this step ensures that no possible candidates for rational roots are overlooked.

Testing Rational Roots

Once potential rational roots are identified using the Rational Root Theorem, each one must be tested to see if it satisfies the equation. Testing involves substituting each candidate \( \frac{p}{q} \) into the polynomial equation. For our case:

• Substitute candidates: \( ±1, ±2, ±4, ±8, ±\frac{1}{2}, ±\frac{1}{4} \).
• Verify by plugging each into the equation: \( 4x^5 - x^4 - x^3 - x^2 + x - 8 = 0 \).
• If substitution results in zero, that candidate is a rational root; if not, it isn’t.

After trying each possible rational root and not achieving a zero, we can confidently conclude that the equation has no rational roots. This method ensures a thorough examination of all possible rational solutions.