Problem 60 - College Algebra

Question

For the following exercises, list all possible rational zeros for the functions. \(f(x)=4 x^{5}-10 x^{4}+8 x^{3}+x^{2}-8\)

Step-by-Step Solution

Answer

Possible rational zeros are \( \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{4} \).

Step 1: Identify the Rational Zero Theorem

The Rational Zero Theorem states that any rational solution, or zero, of the polynomial equation \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \) is 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 2: Determine Leading Coefficient and Constant Term

For the polynomial \( f(x) = 4x^5 - 10x^4 + 8x^3 + x^2 - 8 \), the leading coefficient \( a_n \) is 4 and the constant term \( a_0 \) is -8.

Step 3: Find Factors of Constant Term

The factors of the constant term -8 are \( \pm 1, \pm 2, \pm 4, \pm 8 \).

Step 4: Find Factors of Leading Coefficient

The factors of the leading coefficient 4 are \( \pm 1, \pm 2, \pm 4 \).

Step 5: List Possible Rational Zeros

Using the factors of -8 and 4, list all possible rational zeros as \( \frac{p}{q} \) where \( p \) is a factor of -8 and \( q \) is a factor of 4. This results in the potential rational zeros: \[ \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{2}{4} \]_simplified: \( \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{4} \)

Key Concepts

Polynomial Functions

Polynomial functions are foundational concepts in algebra, characterized by equations composed of terms, each consisting of a variable raised to an exponent and multiplied by a coefficient. Understanding polynomial functions begins with recognizing their general form:

• A polynomial function of degree "n" has the form: \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where each coefficient \(a_i\) is a real number, and \(a_n eq 0\).
• The highest power of the variable in the function, "n," determines the degree of the polynomial.
• The leading coefficient is the coefficient of the term with the highest degree, and the constant term is the term without the variable.

Recognizing these components helps in manipulating and solving polynomial functions. Their graphs provide visual insights into the nature of the function, depicting characteristics like turning points and zeros.

Rational Zero Theorem

The Rational Zero Theorem is a handy tool when determining the possible rational roots of a polynomial function. This theorem posits that any potential rational zero, or root, of the polynomial function can be expressed as \( \frac{p}{q} \), where:

• \( p \) is a factor of the constant term \( a_0 \).
• \( q \) is a factor of the leading coefficient \( a_n \).

Using the Rational Zero Theorem effectively reduces the number of potential solutions one must check when searching for rational zeros. For example, if you have a polynomial \( f(x) = 4x^5 - 10x^4 + 8x^3 + x^2 - 8 \), the constant term is \(-8\) and the leading coefficient is \(4\). This information guides us in listing out all possible rational zeros by considering the ratios of factors of \(-8\) to those of \(4\).

Factors of Coefficients

To apply the Rational Zero Theorem, one must determine the factors of both the constant term and the leading coefficient. Here's how you break it down:

• Identify the constant term \( a_0 \) and find all of its integer factors. For our example, the constant term is \(-8\), and its factors are \( \pm 1, \pm 2, \pm 4, \pm 8 \).
• Similarly, identify the leading coefficient \( a_n \) in the polynomial and list all of its factors. In the same polynomial function, the leading coefficient is \(4\), with factors \( \pm 1, \pm 2, \pm 4 \).
• With these lists, express each possible rational zero as \( \frac{p}{q} \), where \( p \) stems from the factors of the constant term and \( q \) originates from the factors of the leading coefficient.
• For the example polynomial, the potential rational zeros include: \( \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{4} \).

This systematic approach simplifies the search for real solutions to polynomial equations by narrowing down the possibilities to a manageable number.