Problem 1 - Algebra 2

Question

List all of the possible rational zeros of each function. \(p(x)=x^{4}-10\)

Step-by-Step Solution

Answer

The possible rational zeros are ±1, ±2, ±5, ±10.

Step 1: Identify the Leading Coefficient and the Constant Term

The polynomial function given is \( p(x) = x^4 - 10 \). Identify the leading coefficient, which is the coefficient of the highest degree term, and the constant term. In this case, the leading coefficient is 1 and the constant term is -10.

Step 2: Apply the Rational Root Theorem

The Rational Root Theorem states that any rational zero of the polynomial \( p(x) = a_n x^n + \, ... \, + a_0 \) will be of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term \( a_0 \), and \( q \) is a factor of the leading coefficient \( a_n \).

Step 3: List Factors of the Constant Term

List all of the factors of the constant term \(-10\). These factors are \( \pm 1, \pm 2, \pm 5, \pm 10 \).

Step 4: List Factors of the Leading Coefficient

List all of the factors of the leading coefficient, which is 1. The only factors are \( \pm 1 \).

Step 5: Determine Possible Rational Zeros

Using the form \( \frac{p}{q} \), list all possible rational zeros by dividing each factor of the constant term by each factor of the leading coefficient. The possible rational zeros are \( \pm 1, \pm 2, \pm 5, \pm 10 \).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression consisting of variables, constants, and exponents, which can be combined using addition, subtraction, multiplication, and non-negative integer exponents. In a simple sense, it's like a combination of numbers and letters raised to the power. An example of a polynomial function is given as \( p(x) = x^4 - 10 \). Here, the function consists of a single variable \( x \) and a constant term.

• The degree of a polynomial is determined by the highest exponent of its variable, which in this example is 4, indicating it is a fourth-degree polynomial.
• Every term in the polynomial can be represented as \( a_n x^n \) where \( a_n \) is a constant coefficient and \( x \) is a variable of degree \( n \).

Polynomials are essential in solving a variety of mathematical problems and play a key role in algebra.

Leading Coefficient

The leading coefficient is the number found in front of the term with the highest power in a polynomial function. It provides important information about the behavior of the polynomial's graph.
For the function \( p(x) = x^4 - 10 \), the leading term is \( x^4 \). The coefficient of this term is 1, making it the leading coefficient.

• The leading coefficient can affect the width and direction of the graph of a polynomial.
• When the leading coefficient is positive, the ends of the polynomial’s graph in higher power point upwards, and if it’s negative, they point downwards.
• A leading coefficient of 1, like in this polynomial, indicates the simplest form of the polynomial's leading term.

Constant Term

The constant term in a polynomial is the term that does not contain any variables. It stands alone and does not change as the variable changes.
In the polynomial \( p(x) = x^4 - 10 \), the constant term is \(-10\). It is the term without an \( x \) following it.

• Constant terms are integral in determining possible rational zeros using the Rational Root Theorem.
• The term can represent a vertical shift on the graph of the polynomial.
• In equations, constant terms are crucial in solving for specific values as they define the y-intercept when graphed.

Factors of a Number

Factors of a number are integers that can be multiplied together to form the original number. They play a fundamental role in the Rational Root Theorem, which helps determine possible rational zeros of a polynomial.
For example, considering the constant term \(-10\), it has several factors:

• Positive factors: \( 1, 2, 5, 10 \)
• Negative factors: \( -1, -2, -5, -10 \)

In the context of the polynomial \( p(x) = x^4 - 10 \), these factors help us determine which rational zeros might exist by dividing them by the factors of the leading coefficient. It's vital to consider both positive and negative numbers as factors, doubling the potential rational roots.