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

Step-by-Step Solution

Answer

The possible rational zeros are \(\pm 1\) and \(\pm 3\).

Step 1: Identify the Constant Term and Leading Coefficient

The Rational Zeros Theorem suggests that potential rational zeros of a polynomial can be found as ratios of the factors of the constant term to the factors of the leading coefficient. For the polynomial \(P(x) = x^3 - 4x^2 + 3\), the constant term is 3 and the leading coefficient of \(x^3\) is 1.

Step 2: List the Factors of the Constant Term

The constant term of the polynomial is 3. The factors of 3 are \(\pm 1\) and \(\pm 3\). These will serve as the numerators in our potential rational zeros.

Step 3: List the Factors of the Leading Coefficient

The leading coefficient is 1. The factors of 1 are \(\pm 1\). These will serve as the denominators in our potential rational zeros.

Step 4: Find All Possible Ratios

To find all possible rational zeros, divide each of the factors of the constant term by each of the factors of the leading coefficient. The potential rational zeros are given by:\[\frac{\pm 1}{\pm 1}, \frac{\pm 3}{\pm 1}\]Simplifying these ratios gives \(\pm 1, \pm 3\).

Key Concepts

Polynomials

Polynomials are mathematical expressions composed of variables and constants, combined using addition, subtraction, multiplication, and non-negative integer exponents. They form a significant part of algebra as they describe curves and solve equations.

In our context, when looking at the polynomial \(P(x) = x^3 - 4x^2 + 3\), we observe a third-degree polynomial. This means that the highest power of the variable \(x\) is three, classifying it as a cubic polynomial.

• The degrees of polynomials determine the general shape of their graphs.
• Higher degree polynomials usually have more complex graphs and more roots (including rational, irrational, and complex roots).

Understanding polynomials is crucial as they are the foundation for more advanced mathematical concepts and applications.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It plays a vital role in determining the behavior of the polynomial, especially as the variable \(x\) approaches very large or small values.

In the polynomial \(P(x) = x^3 - 4x^2 + 3\), the leading term is \(x^3\), and its coefficient is 1.

• One key property is that the sign and magnitude of the leading coefficient can affect the end behavior of the polynomial's graph.
• If the leading coefficient is positive, as \(x\) approaches positive or negative infinity, the graph will rise, assuming the degree of the polynomial is odd.

Constant Term

The constant term is the term in the polynomial that does not contain any variables. It's found by looking at the zero-degree term. In a polynomial like \(P(x) = x^3 - 4x^2 + 3\), the constant term is 3, which affects the vertical position of the graph on the y-axis when \(x = 0\).

• In the Rational Zeros Theorem, the constant term’s factors are crucial as they contribute to the list of potential numerators for rational roots.
• The position of the graph relative to the x-axis can be partially determined by the constant term.

The constant term's role in graph behavior and potential zeros makes it an essential component of polynomial analysis.

Factors

Factors are numbers or expressions that divide another number or expression evenly—without leaving a remainder. In polynomials, we often focus on finding the factors of the leading coefficient and the constant term to apply the Rational Zeros Theorem.

For the polynomial \(P(x) = x^3 - 4x^2 + 3\):

• The factors of 3 (constant term) are \(\pm 1\) and \(\pm 3\).
• The factors of 1 (leading coefficient) are \(\pm 1\).

By forming ratios of these factors, the Rational Zeros Theorem guides us to possible rational roots: \(\pm 1\) and \(\pm 3\). This methodology provides a logical approach to identifying and testing potential roots in polynomials.