Problem 57 - College Algebra with Corequisite Support

Question

For the following exercises, list all possible rational zeros for the functions. $$ f(x)=2 x^{3}+3 x^{2}-8 x+5 $$

Step-by-Step Solution

Answer

Possible rational zeros: \( +1, -1, +5, -5, +\frac{1}{2}, -\frac{1}{2}, +\frac{5}{2}, -\frac{5}{2} \).

Step 1: Understanding Rational Root Theorem

The Rational Root Theorem states that for a polynomial \( f(x) = a_n x^n + a_{n-1}x^{n-1} + \, ... \, + a_0 \), any rational root \( \frac{p}{q} \) must have \( p \) as a factor of the constant term \( a_0 \) and \( q \) as a factor of the leading coefficient \( a_n \). This helps us determine the possible rational zeros of the polynomial.

Step 2: Identify the Constant and Leading Coefficient

For the given polynomial \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), the constant term is \( a_0 = 5 \) and the leading coefficient is \( a_n = 2 \). These will help us find all possible rational zeros based on their factors.

Step 3: Find Factors of the Constant Term

The constant term is \( 5 \). The factors of \( 5 \) are \( +1, -1, +5, -5 \). These are the potential numerators \( p \) for the rational roots \( \frac{p}{q} \).

Step 4: Find Factors of the Leading Coefficient

The leading coefficient is \( 2 \). The factors of \( 2 \) are \( +1, -1, +2, -2 \). These are the potential denominators \( q \) for the rational roots \( \frac{p}{q} \).

Step 5: List All Possible Rational Zeros

Using the factors of the constant term and the leading coefficient, form all possible rational numbers \( \frac{p}{q} \). These are:\[ +1, -1, +5, -5, +\frac{1}{2}, -\frac{1}{2}, +\frac{5}{2}, -\frac{5}{2} \]Thus, these are the potential rational zeros of the function \( f(x) \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve a sum of powers in one or more variables multiplied by coefficients. They are a key fundamental concept in algebra. A typical polynomial is in the form:

• \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 \)

This expression is made up of different terms, where each can be written as \( a_ix^i \), with \( i \) being a non-negative integer. Each term has a coefficient \( a_i \), which is a real number, and a variable \( x \) raised to the power of \( i \).
Polynomials can have various degrees. The degree is determined by the highest power of the variable present.
For instance, if the highest power of \( x \) is 3, like in \( 2x^3 + 3x^2 - 8x + 5 \), the polynomial is a cubic function. The degree gives an idea of the polynomial's shape and the number of roots it could have.

Constant Term

The constant term in a polynomial is the term that does not contain any variables, simply a constant value. It is denoted \( a_0 \) in the general expression of a polynomial:

• \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 \)

The constant term is important when applying the Rational Root Theorem, as it influences the potential rational roots.
In the polynomial \( f(x) = 2x^3 + 3x^2 - 8x + 5 \), the constant term is \( 5 \).
This value is critical as it helps in determining the numerators for possible rational zeros. The factors of the constant term (like \( 1, -1, 5, -5 \)) are combined with the factors of the leading coefficient to list all potential rational solutions.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. It is the first non-zero coefficient when a polynomial is written in standard form:

• \( a_nx^n \) in \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 \)

The leading coefficient is essential in analyzing polynomial behavior and plays a crucial role in the Rational Root Theorem.
For the polynomial \( 2x^3 + 3x^2 - 8x + 5 \), the leading coefficient is \( 2 \). This coefficient helps determine the denominators \( q \) for possible rational roots \( \frac{p}{q} \). The factors of the leading coefficient are used with those of the constant term to generate potential rational zeros of the polynomial.

Factors

Factors in mathematics refer to numbers or expressions that multiply to yield a given number or expression. When applying the Rational Root Theorem, factors are used to explore potential rational zeros of a polynomial.
Factors of the constant term are used as potential numerators \( p \), while factors of the leading coefficient are potential denominators \( q \).
Let's go through the process:

• For the polynomial \( 2x^3 + 3x^2 - 8x + 5 \), factors of the constant term \( 5 \) are \( 1, -1, 5, -5 \).
• Factors of the leading coefficient \( 2 \) are \( 1, -1, 2, -2 \).

By forming combinations of these factors as fractions \( \frac{p}{q} \), we get the possible rational roots:

• \( 1, -1, \frac{1}{2}, -\frac{1}{2}, 5, -5, \frac{5}{2}, -\frac{5}{2} \)

These fractions represent all potential rational zeros based on the factors.