Problem 51 - Algebra 2

Question

List all of the possible rational zeros of each function. $$ h(x)=-4 x^{3}-86 x^{2}+57 x+20 $$

Step-by-Step Solution

Answer

Possible rational zeros are \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20, \pm \frac{1}{2}, \pm \frac{5}{2} \).

Step 1: Identify the Coefficients and Constant Term

First, identify the leading coefficient of the polynomial and the constant term. For the polynomial \( h(x) = -4x^3 - 86x^2 + 57x + 20 \), the leading coefficient (\(a_n\)) is \(-4\) and the constant term (\(a_0\)) is \(20\).

Step 2: List the Factors of the Constant Term

List all the integer factors of the constant term \(20\). These are: \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \).

Step 3: List the Factors of the Leading Coefficient

List all the integer factors of the leading coefficient \(-4\). These are: \( \pm 1, \pm 2, \pm 4 \).

Step 4: Formulate Possible Rational Zeros

Using the Rational Root Theorem, the possible rational zeros are all the ratios of the factors of the constant term (Step 2) over the factors of the leading coefficient (Step 3). This gives us: \[ \frac{\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20}{\pm 1, \pm 2, \pm 4} \].

Step 5: Simplify the Ratios

Compute all possible simplified values of the ratios derived in Step 4. These are: \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20, \pm \frac{1}{2}, \pm \frac{5}{2} \).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables, each multiplied by a coefficient. The general form of a polynomial function in one variable, say \( x \), is:

• \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants known as the coefficients, and \( n \) is a non-negative integer representing the degree of the polynomial.
In the given example, the polynomial is \( h(x) = -4x^3 - 86x^2 + 57x + 20 \). This is a cubic polynomial because its highest power of \( x \) is 3. Understanding the structure of polynomial functions helps in solving various algebraic equations and modeling real-world phenomena.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest power. It plays a significant role in determining the end behavior of the polynomial function, especially for those of higher degrees.
In the polynomial \( h(x) = -4x^3 - 86x^2 + 57x + 20 \), the leading coefficient is \(-4\) because it multiplies the term \( x^3 \), which is the highest power in the polynomial.
This coefficient can affect the width and direction (upward or downward) of the graph of the polynomial function. Understanding the influence of the leading coefficient is essential when examining the polynomial's graph.

Constant Term

In a polynomial function, the constant term is the term without a variable. It is found by setting the variable(s) to zero and looking at what remains. For example, in the polynomial \( h(x) = -4x^3 - 86x^2 + 57x + 20 \), the constant term is \( 20 \).
The constant term contributes to the value of the polynomial when the variable is zero. It has no effect on the end behavior or degree of the polynomial but is crucial when using the Rational Root Theorem to determine possible rational zeros.
Understanding the constant term is vital when analyzing polynomial functions and their properties.

Factors

Factors are numbers or expressions that divide another number or expression evenly, meaning without leaving a remainder. When dealing with polynomials, both the constant term and the leading coefficient need to be factored to apply the Rational Root Theorem.
For the polynomial \( h(x) = -4x^3 - 86x^2 + 57x + 20 \):

• The factors of the constant term (20) are: \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \).
• The factors of the leading coefficient (-4) are: \( \pm 1, \pm 2, \pm 4 \).

These factors help determine the possible rational zeros of the polynomial by being combined into ratios.

Rational Zeros

Rational zeros are the potential roots of a polynomial that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers with no common factors other than 1, and \( q \) is not zero. The Rational Root Theorem provides a way to list these possible zeros of a polynomial.
The theorem states that for a polynomial function \( a_nx^n + \ldots + a_0 \), any rational solution must be of the form \( \frac{\text{factor of } a_0}{\text{factor of } a_n} \).
In our polynomial \( h(x) = -4x^3 - 86x^2 + 57x + 20 \), the possible rational zeros are:

• \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20, \pm \frac{1}{2}, \pm \frac{5}{2} \).

Simplifying these fractions gives the potential solutions to the equation \( h(x) = 0 \), helping to find any actual rational roots efficiently.