Problem 13 - Precalculus with Calculus Previews

Question

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=x^{5}+x^{4}-5 x^{3}+x^{2}-6 x $$

Step-by-Step Solution

Answer

The rational zeros are \(x = 0, 1, -2\).

Step 1: Identify the polynomial function

The given polynomial function is \( f(x) = x^5 + x^4 - 5x^3 + x^2 - 6x \) and it is a polynomial of degree 5.

Step 2: Write the Rational Root Theorem

According to the Rational Root Theorem, any rational root of the polynomial \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 \) must be of the form \( \frac{p}{q} \), where \( p \) is a factor of \( a_0 \) (the constant term), and \( q \) is a factor of \( a_n \) (the leading coefficient).

Step 3: Identify factors of constant and leading coefficient

Here, the leading coefficient \( a_n = 1 \) and the constant term is \( a_0 = 0 \), however the actual polynomial is \( f(x) = x(x^4 + x^3 - 5x^2 + x - 6) \). Since \( x^5 \) has no constant term, we can consider factors for \( x(x^4 + x^3 - 5x^2 + x - 6) \) which has \( -6 \) as its motivating constant factor.

Step 4: List all possible rational zeros

The possible rational zeros of the function are all factors of \(-6\), since the leading coefficient is 1. The factors of \(-6\) are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Step 5: Test each candidate rational root

Substitute each possible factor into the polynomial \( f(x) = x^4 + x^3 - 5x^2 + x - 6 \) to test for zero values.1. \( f(x) = 0 \) when \( x = 1 \), thus \( x-1 \) is a factor.2. Try other numbers and find:\( x = -2 \) also gives \( f(-2) = 0 \), so \( x+2 \) is another factor.

Step 6: Conclude with the rational zeros found

After testing, the rational roots of \( f(x) = x(x^4 + x^3 - 5x^2 + x - 6) \) are \( x = 0, x = 1, x = -2 \).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For example, the function given in our exercise is \( f(x) = x^5 + x^4 - 5x^3 + x^2 - 6x \). This function is a single-variable polynomial because it involves only the variable \( x \).

Each term in a polynomial has a coefficient (for instance, 1 in \( x^5 \), or -5 in \( -5x^3 \)) and a power of the variable. These elements make polynomials versatile tools in representing and solving mathematical problems.

Polynomials can be classified according to their degree, which is determined by the highest power of the variable (more on that in the third section). Polynomials are fundamental not just in algebra but also in calculus, as they form the basis for many approximations and functions.

Rational Zeros

Rational zeros of a polynomial are the zeros that can be expressed as a fraction of two integers, \( \frac{p}{q} \), where \( p \) and \( q \) are factors of the constant term and leading coefficient respectively. Understanding this concept allows us to apply the Rational Root Theorem effectively.

In our given polynomial \( f(x) = x(x^4 + x^3 - 5x^2 + x - 6) \), we are specifically looking at the expression \( x^4 + x^3 - 5x^2 + x - 6 \). Here, it's crucial to remember that the constant term in the most reduced form of the expression (without the factor \( x \)) is \( -6 \) and the leading coefficient is 1.

According to the Rational Root Theorem, we generate possible rational zeros by combining the factors of the constant term with the factors of the leading coefficient. Thus, for \( f(x) \), the rational zeros candidates are \( \pm 1, \pm 2, \pm 3, \pm 6 \). Testing these values in the polynomial helps us find the actual zeros, simplifying our problem-solving process.

Degree of Polynomial

The degree of a polynomial is the largest exponent of the variable in the function. This degree offers insights into the function's behavior, such as the number of roots the polynomial can have, including complex and real roots.

For the polynomial \( f(x) = x^5 + x^4 - 5x^3 + x^2 - 6x \), the highest degree is 5 because the highest power of \( x \) is \( x^5 \). This tells us that in theory, this polynomial can have up to 5 roots. These roots can be a combination of real and complex numbers.

In this particular exercise, we identified the rational zeros to be \( x = 0, x = 1, \) and \( x = -2 \). While these are just three of the possible roots, other roots might exist as irrational or complex numbers. Understanding the degree is essential as it helps predict the maximum number of times the curve can intersect the x-axis.