Problem 72 - Algebra and Trigonometry

Question

List all potential rational zeros of \(P(x)=2 x^{3}-5 x^{2}+13 x+6\)

Step-by-Step Solution

Answer

The potential rational zeros are: \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \).

Step 1 - Identify Constants

Identify the constant term and the leading coefficient of the polynomial. For the polynomial \(P(x) = 2x^3 - 5x^2 + 13x + 6\), the constant term \((a_0)\) is 6, and the leading coefficient \((a_n)\) is 2.

Step 2 - Determine Factors of the Constant Term

List all factors of the constant term 6. The factors are: \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Step 3 - Determine Factors of the Leading Coefficient

List all factors of the leading coefficient 2. The factors are: \( \pm 1, \pm 2 \).

Step 4 - Form Possible Rational Zeros

Form all possible rational zeros using the factors of the constant term over the factors of the leading coefficient. Using the Rational Root Theorem: \( \text{Possible zeros} = \frac{\text{factors of } a_0}{\text{factors of } a_n} \). This results in all possible combinations: \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \).

Key Concepts

Polynomial Functions

A polynomial function is a type of mathematical expression that involves a sum of terms, each consisting of a variable raised to a whole number exponent, and each multiplied by a constant coefficient. For instance, the polynomial function in our example is:

P(x) = 2x^3 - 5x^2 + 13x + 6

Here, the terms are combined to form the polynomial. Polynomial functions are significant in algebra and calculus because they are used to model a variety of real-world situations. They can represent anything from simple growth patterns to the more complex behaviors found in physics and engineering. Understanding their properties is essential for solving many mathematical problems.

Factors of Constants

To find the potential rational zeros of a polynomial, we need to examine the factors of the constant term. The constant term is the term in the polynomial that does not include any variables. For our polynomial function, P(x) = 2x^3 - 5x^2 + 13x + 6,

This makes the constant term (a_0) equal to 6.

We then list all the factors of 6. Factors are numbers that can be multiplied together to get 6. These include the positive and negative integers that satisfy this property:

• ±1
• ±2
• ±3
• ±6

This list of factors will be useful in forming the possible rational zeros.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest power in a polynomial. In our polynomial, P(x) = 2x^3 - 5x^2 + 13x + 6

The term with the highest power is 2x^3, and its leading coefficient (a_n) is 2.

Determining the factors of the leading coefficient is a crucial step in the Rational Zero Theorem, which helps us generate possible rational zeros. We list all factors of 2:

• ±1
• ±2

These factors will create combinations with the factors of the constant term to form potential rational zeros.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations. The Rational Zero Theorem is an important tool in algebra that helps us find all possible rational zeros of a polynomial function.

The theorem states that every rational zero of a polynomial function will be a fraction formed by the factors of the constant term (a_0) divided by the factors of the leading coefficient (a_n).

Applying the theorem to our polynomial, we combine the factors of 6 and 2 to form potential zeros:

• ±1
• ±2
• ±3
• ±6
• ±1/2
• ±3/2

This list represents all possible rational zeros of the polynomial. Algebra techniques, including the Rational Zero Theorem, are powerful tools for solving diverse mathematical problems.