Problem 8 - Algebra and Trigonometry

Question

List all possible rational zeros given by the Rational Zeros Theorem (but don' t check to see which actually are zeros). $$ S(x)=6 x^{4}-x^{2}+2 x+12 $$

Step-by-Step Solution

Answer

Possible rational zeros: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm \frac{3}{2}, \pm 4, \pm \frac{4}{3}, \pm 6, \pm 12 \).

Step 1: Understand the Rational Zeros Theorem

The Rational Zeros Theorem states that if a polynomial has rational zeros, they are of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.

Step 2: Identify the Constant and Leading Coefficient

For the polynomial \( S(x) = 6x^4 - x^2 + 2x + 12 \), the constant term is 12, and the leading coefficient is 6. These will be used to find factors \( p \) and \( q \).

Step 3: Determine the Factors of the Constant Term

The factors of the constant term 12 are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 4: Determine the Factors of the Leading Coefficient

The factors of the leading coefficient 6 are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Step 5: List Possible Rational Zeros

Combine the factors from Step 3 and Step 4 to generate possible rational zeros. Form the fractions \( \frac{p}{q} \) using each factor of 12 over each factor of 6: \[ \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm \frac{3}{2}, \pm 4, \pm \frac{4}{3}, \pm 6, \pm 12 \]. These are the potential rational zeros of the polynomial.

Key Concepts

Polynomial

A polynomial is like a mathematical sentence involving numbers and variables combined through addition, subtraction, and multiplication.
Each term in a polynomial is made up of a coefficient and a variable raised to a whole number power. The variable could be any letter, commonly x or y.
In our exercise, we looked at a polynomial expression, specifically:

• The highest power of the variable, 4, makes it a fourth-degree polynomial.
• This polynomial is written as: \( S(x) = 6x^4 - x^2 + 2x + 12 \).

Each part, like \( 6x^4 \), is a term. Understanding these parts helps us work with the concepts in the Rational Zeros Theorem.

Factors

Factors are numbers you multiply to get another number. In polynomials, the process of factoring involves breaking down an expression into a product of simpler polynomials.
For the Rational Zeros Theorem, factors play a critical role.

• Each non-zero constant and coefficient has factors.
• For instance, the number 12 can be broken down into its factors: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).
• Similarly, the number 6 can be broken down into: \( \pm 1, \pm 2, \pm 3, \pm 6 \).

These factors help us form potential rational zeros by dividing the factors of the constant term by the factors of the leading coefficient.

Leading Coefficient

The leading coefficient of a polynomial is the number in front of the highest power of the variable.
This number is significant because, in the Rational Zeros Theorem, its factors are used to help determine possible rational zeros.In our polynomial example:

• The term with the highest power is \( 6x^4 \).
• Here, 6 is the leading coefficient.

When considered with the factors of the constant term, the factors of the leading coefficient help generate a complete list of potential rational solutions for a polynomial equation.

Constant Term

In a polynomial, the constant term is the term without a variable.
It is just a number that remains unchanged, regardless of the value of the variable.For example with \( S(x) = 6x^4 - x^2 + 2x + 12 \), the constant term is 12.
This number is important because its factors are used in the Rational Zeros Theorem.

• The factors of the constant term, like 12 in our example, generate possible values for \( p \) in the Rational Zeros Theorem.
• This helps form the fractions \( \frac{p}{q} \) where \( p \) is a factor of the constant term.

By using the constant term and combining it with other factors, we can predict possible rational zeros of a polynomial.