Problem 3 - 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). $$ R(x)=2 x^{5}+3 x^{3}+4 x^{2}-8 $$

Step-by-Step Solution

Answer

Possible rational zeros: \( \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2} \).

Step 1: Identify Factors of Constant Term

The Rational Zeros Theorem states that if \( R(x) = a_n x^n + \.\.\. + a_0 \) has a rational zero \( \frac{p}{q} \), then \( p \) is a factor of the constant term \( a_0 \). Here, \( a_0 = -8 \). The factors of \(-8\) are \( \pm 1, \pm 2, \pm 4, \pm 8 \).

Step 2: Identify Factors of Leading Coefficient

According to the theorem, \( q \) is a factor of the leading coefficient \( a_n \). Here, \( a_n = 2 \). The factors of \(2\) are \( \pm 1, \pm 2 \).

Step 3: Formulate Possible Rational Zeros

The possible rational zeros are given by \( \frac{p}{q} \) for each combination of the factors of \( a_0 \) and \( a_n \). This gives us the following possible rational zeros: \( \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{2}{2} (= \pm 1), \pm \frac{4}{2} (= \pm 2), \pm \frac{8}{2} (= \pm 4) \). Thus, the complete list of different possible rational zeros is \( \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2} \).

Key Concepts

Polynomial Roots

A polynomial root refers to any value of the variable for which a polynomial equation becomes zero. Finding the roots of a polynomial is crucial because these roots represent the solutions to the equation. For example, if we have a polynomial \( R(x) = 2x^5 + 3x^3 + 4x^2 - 8 \), then the roots of this polynomial are the values of \( x \) that satisfy \( R(x) = 0 \). Calculating these roots, especially rational ones, can help us understand the behavior of the polynomial graphically and algebraically.
To find rational roots, one useful strategy is to apply the Rational Zeros Theorem, which can simplify this task by allowing us to generate a list of potential candidates for the roots rather than analyzing every single number. Rational roots can be particularly important not just for solving equations but for understanding and graphing polynomial functions. When these polynomials are plotted, the roots represent the values where the graph intercepts the x-axis. Therefore, identifying rational roots simplifies forecasting where a polynomial's graph touches the x-axis.

Constant Term Factors

The constant term in a polynomial is the term without any variables; it lies at the end of the polynomial expression. In our given polynomial \( R(x) = 2x^5 + 3x^3 + 4x^2 - 8 \), the constant term is \(-8\). Determining the factors of the constant term is a critical step when utilizing the Rational Zeros Theorem, which guides us to potential rational zeros of the polynomial.

• Factors of \(-8\) are \( \pm 1, \pm 2, \pm 4, \pm 8 \).

These factors (\( p \)) are placed into fraction form with the factors of the polynomial's leading coefficient, allowing us to ascertain possible rational combinations that may result in polynomial zeros. Recognizing these factors in a polynomial problem helps narrow down potential zero candidates, making it much easier to subsequently verify which of these are actual roots.

Leading Coefficient Factors

The leading coefficient of a polynomial is the coefficient of its highest degree term. In \( R(x) = 2x^5 + 3x^3 + 4x^2 - 8 \), the leading term is \( 2x^5 \), hence the leading coefficient is \(2\). When applying the Rational Zeros Theorem, finding the leading coefficient's factors is essential because they constitute the denominator \( q \) in potential rational zero forms \( \frac{p}{q} \).

• For the leading coefficient \(2\), factors are \( \pm 1, \pm 2 \).

Using these factors, you can calculate possible rational zeros systematically. Combine these factors with those from the constant term, \(-8\). This systematic pairing ensures we consider every possible simple fraction formed from tying each factor of the constant term with each of the leading coefficient's factors. Pairing them generates candidates like \( \pm 1, \pm \frac{1}{2}, \pm 4, \) and so on. The Rational Zeros Theorem allows one to efficiently reduce the otherwise cumbersome task of finding all potential rational solutions.