Question

Describe and correct the error in listing the possible rational zeros of the function. $$ f(x)=x^3+5 x^2-9 x-45 $$ Possible rational zeros of \(f\) : $$ 1,3,5,9,15,45 $$

Short answer

The error in the list is the inclusion of the numbers 3 and 9. So, the corrected list of possible rational zeros of the function \( f \) should be ±1, ±5, ±15, ±45.

Step-by-step solution

Understand the Rational Root Theorem

Rational Root Theorem states that if \( p/q \) is a rational root of a polynomial equation where \( p \) is a factor of the constant term (in this case -45) and \( q \) is a factor of the leading coefficient (in this case 1).

Identify the Error

In the given list of rational zeros of \( f \), the number 3 and 9 do not actually divide the constant term -45, so it’s incorrectly listed.

Correcting the List

The correct list should only include factors of -45, which are ±1, ±3, ±5, ±9, ±15, ±45. So the correct list of possible rational zeros of \( f \) should be ±1, ±5, ±15, ±45.

Key concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. A typical polynomial function can be written in the form:
\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0 \]
where each term consists of a coefficient (like the numbers \(a_n\), \(a_{n-1}\), etc.), multiplied by the variable \(x\) raised to a non-negative integer power. The highest power of \(x\) in the polynomial is referred to as the degree of the polynomial, which determines many of the function's properties, including the number of zeros it may have.
Polynomial functions have various applications across physics, engineering, and economics due to their simplicity and the ease of calculating their derivatives and integrals. They can be graphed to show their roots or zeros, which are the values of \(x\) for which the function equals zero. Determining these zeros is important for solving polynomial equations and modeling real-world scenarios.

Characteristics of Polynomial Functions

• They are continuous and smooth curves.
• They can have multiple zeros.
• The number of turning points is at most the degree of the polynomial minus one.
• Depending on the degree and the leading coefficient, the ends of the polynomial graph can extend indefinitely upwards or downwards.

Rational Root Theorem

The Rational Root Theorem, also known as the Rational Zero Theorem, is a useful tool to determine the potential rational zeros of a polynomial function. It states that if a polynomial function has rational zeros (or roots), they can be expressed in the form \( \frac{p}{q} \), where \(p\) and \(q\) have the following characteristics:

• \(p\) is an integer factor of the constant term at the end of the polynomial (\(a_0\)).
• \(q\) is an integer factor of the leading coefficient (\(a_n\)).

To use the theorem effectively, one would list out all possible factors of \(a_0\) and \(a_n\). Every combination of these factors (positive and negative) may be a potential zero of the polynomial. It is important to note that while the theorem can give you a list of potential rational zeros, not all listed values will necessarily be actual zeros of the function—it merely narrows down the possibilities for further testing.

Applying the Rational Root Theorem

Following the theorem, for the polynomial function \( f(x) = x^3 + 5x^2 - 9x - 45 \), the constant term is -45, and the leading coefficient is 1. Thus, the list of possible rational zeros should be based on the factors of -45 alone since the leading coefficient is 1 (making all factors of 1 trivially just ±1). From the original exercise, the error occurred because not all factors of -45 were considered, and more importantly, the list did not account for the negative factors, which can equally serve as potential zeros.

Factoring Polynomials

Factoring polynomials is an essential skill for simplifying polynomial expressions and solving polynomial equations. To factor a polynomial is to express it as the product of two or more polynomials of lesser degree. These factors often reveal the zeros of the polynomial by setting each factor equal to zero and solving for the variable.
There are various methods for factoring polynomials:

• Factoring by grouping.
• Using the square of a binomial.
• Applying the difference of squares.
• Utilizing the sum or difference of cubes.
• Employing the quadratic formula or completing the square for quadratic equations.

It is also important to look for a greatest common factor (GCF) before attempting other methods, as this can greatly simplify the process. Once you have factored the polynomial, you can set each factor equal to zero to find the polynomial's zeros, aligning with the solutions provided by the Rational Root Theorem when applicable. With the correct approach and a careful application of these methods, most polynomials can be broken down into more manageable pieces and solved effectively.

Challenges in Factoring

While many polynomials, especially quadratics, have textbook methods for factoring, higher-degree polynomials may present more of a challenge and require more creative strategies. Combining several factoring methods, understanding the role of complex numbers, and having familiarity with synthetic division or long division of polynomials greatly enhance one's ability to master this crucial area of algebra.