Problem 4 - 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 are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2} \).

Step 1: Identify coefficients of polynomial

The polynomial given is \( S(x) = 6x^4 - x^2 + 2x + 12 \). Identify the leading coefficient \( a_n \) and the constant term. Here, \( a_n = 6 \) and the constant term \( a_0 = 12 \).

Step 2: Determine factors of the constant term

List all possible integer factors of the constant term, \( a_0 = 12 \). These factors are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 3: Determine factors of the leading coefficient

List all possible integer factors of the leading coefficient, \( a_n = 6 \). These factors are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Step 4: List all possible rational zeros using factors

According to the Rational Zeros Theorem, possible rational zeros are given by \( \frac{p}{q} \) where \( p \) is a factor of \( a_0 \) and \( q \) is a factor of \( a_n \). Form all possible fractions \( \frac{p}{q} \) using the factors of 12 and the factors of 6. The possible rational zeros are: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2} \).

Key Concepts

Polynomial Functions

Understanding polynomial functions is crucial when studying algebra. A polynomial function, such as the function given in the exercise, is an expression composed of variables, coefficients, and non-negative integer exponents. It is a sum of terms, where each term consists of a coefficient multiplied by a variable raised to a power.

For example, in the polynomial function \( S(x) = 6x^4 - x^2 + 2x + 12 \), each term has its own degree, which is the power of \( x \) in that term:

• \( 6x^4 \) is the highest degree term and is what makes this a fourth-degree polynomial.
• \( -x^2 \), \( 2x \), and \( 12 \) are subsequent terms with decreasing powers of \( x \).

Polynomial functions are important because they appear in various mathematical contexts and can represent a wide range of real-world phenomena. Understanding their structure helps you effectively analyze and solve different equations.

Integer Factors

Integer factors play a vital role when applying the Rational Zeros Theorem. An integer factor of a number is an integer that can be multiplied by another integer to give the original number. For example, the integer factors of 12 are any numbers that divide 12 evenly:

• Positive factors: 1, 2, 3, 4, 6, and 12.
• Negative factors: -1, -2, -3, -4, -6, and -12.

Both positive and negative integer factors are essential when calculating possible rational zeros. The process involves identifying the factors of both the leading coefficient and the constant term of a polynomial.

Understanding integer factors allows you to list all potential numerators and denominators that may result in a rational zero, according to the Rational Zeros Theorem. This list lays the foundation for examining the behavior of polynomial functions further.

Rational Zeros

The Rational Zeros Theorem is a powerful tool that helps determine all possible rational zeros of a polynomial function. A rational zero is a solution to the equation formed by setting the polynomial equal to zero.

To find potential rational zeros, you'll calculate all possible values using the formula \( \frac{p}{q} \), where:

• \( p \) is a factor of the constant term of the polynomial service function.
• \( q \) is a factor of the leading coefficient.

By generating all combinations of \( \frac{p}{q} \), the theorem provides a list of potential candidates that can be checked to identify actual zeros.

It's important to note that these possible rational zeros need to be verified by substituting back into the polynomial to confirm which ones are actual zeros. The theorem simplifies finding zeros by focusing your efforts on a specific set of rational numbers.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It tells us a lot about the polynomial's graph and how to apply the Rational Zeros Theorem effectively.

In the polynomial function \( S(x) = 6x^4 - x^2 + 2x + 12 \), the leading coefficient is the number 6.

The significance of identifying the leading coefficient is multifaceted:

• It helps in the application of the Rational Zeros Theorem by determining possible values of \( q \) in the formulation \( \frac{p}{q} \).
• It affects the end behavior of the polynomial's graph.
• A high leading coefficient generally indicates that the polynomial grows or shrinks more rapidly as \( x \) moves away from zero.

By understanding the leading coefficient's role, you gain insights into both the algebraic properties and the graphical representation of polynomial functions. This understanding is especially useful in higher-level mathematics and practical applications.