Problem 5 - Precalculus Mathematics for Calculus

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

Possible rational zeros are \(\pm1, \pm3\).

Step 1: Identify the Constant Term

The constant term of the polynomial \(P(x) = x^3 - 4x^2 + 3\) is \(3\).

Step 2: Identify the Leading Coefficient

The leading coefficient of the polynomial \(P(x) = x^3 - 4x^2 + 3\) is \(1\) because the coefficient of \(x^3\) is \(1\).

Step 3: List the Factors of the Constant Term

The factors of the constant term \(3\) are \(1, -1, 3,\) and \(-3\).

Step 4: List the Factors of the Leading Coefficient

The factors of the leading coefficient \(1\) are \(1\) and \(-1\).

Step 5: Determine Possible Rational Zeros

According to the Rational Zeros Theorem, the possible rational zeros are given by \(\pm\frac{\text{factors of the constant term}}{\text{factors of the leading coefficient}}\). Thus, the possible rational zeros are \(\pm1, \pm3\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number exponents. A polynomial is composed of one or more terms, each of which includes a constant multiplied by a variable raised to a power. For example, in the polynomial function \( P(x) = x^3 - 4x^2 + 3 \):

• The term \( x^3 \) is a monomial where the variable \( x \) is raised to the power of three, and its coefficient is 1.
• The term \( -4x^2 \) has the variable raised to the power of two, with a coefficient of -4.
• The term \( 3 \) is a constant term because it does not contain a variable.

Polynomials are defined based on the highest power, known as the degree of the polynomial. Here, the degree of this polynomial is 3 because the highest power of the variable \( x \) is 3. Understanding polynomial functions is fundamental as they appear frequently in algebra, calculus, and apply to many real-world scenarios where relationships between quantities can be modeled mathematically.

Factors of Integers

Factors of integers play a crucial role in determining the possible rational zeros of a polynomial. Factors of an integer are numbers that divide the integer evenly without leaving a remainder. Let's focus on the polynomial \( P(x) = x^3 - 4x^2 + 3 \) and how factors come into play:

• The constant term (last term without variable) is 3. The factors of 3 are the integers that multiply together to give 3: 1, -1, 3, and -3.

To find these factors:

• Consider both positive and negative values because a negative times a negative equals positive.
• Always start with 1 and -1; they are factors of every integer.

The factors are essential for applying the Rational Zeros Theorem, as they help in listing all potential rational zeros, which are simplified forms of fractions where the numerator comes from the factors of the constant term.

Leading Coefficient

The leading coefficient is an important concept in polynomial functions. It is the coefficient of the term with the highest power of the variable in the polynomial. For the given polynomial \( P(x) = x^3 - 4x^2 + 3 \):

• The term \( x^3 \) is the highest degree term, and its coefficient is 1, making it the leading coefficient.

The leading coefficient influences the shape and direction of the polynomial graph. In our solution, it aids in calculating the possible rational zeros by being in the denominator. According to the Rational Zeros Theorem:

• Divide factors of the constant term 3 by factors of the leading coefficient 1, to find possible zeros: \( \pm\frac{3}{1}, \pm\frac{1}{1} \).

Despite being 1 in this scenario, which simplifies computations, a leading coefficient different from 1 would greatly affect the divisor list and, thus, all potential zeros derived. Understanding the leading coefficient is crucial for advanced polynomial analysis.