Question

List the potential rational zeros of each polynomial function. Do not attempt to find the zeros. $$ f(x)=-6 x^{3}-x^{2}+x+10 $$

Short answer

The potential rational zeros are the fractions \(\frac{\text{factors of 10}}{\text{factors of -6}}\).

Step-by-step solution

Identify the Leading Coefficient

The leading coefficient of the polynomial function \(f(x) = -6x^3 - x^2 + x + 10\) is the coefficient of the term with the highest degree, which is -6.

Identify the Constant Term

The constant term of the polynomial function \(f(x) = -6x^3 - x^2 + x + 10\) is the term without any variable, which is 10.

List the Factors of the Constant Term

The factors of the constant term (10) are \(\boxed{\text{-10, -5, -2, -1, 1, 2, 5, 10}}\).

List the Factors of the Leading Coefficient

The factors of the leading coefficient (-6) are \(\boxed{\text{-6, -3, -2, -1, 1, 2, 3, 6}}\).

Write the Potential Rational Zeros

The potential rational zeros are the ratios of the factors of the constant term to the factors of the leading coefficient. List all possible combinations: \(\frac{10}{6}, \frac{10}{3}, \frac{10}{2}, \frac{10}{1}, \frac{5}{6}, \frac{5}{3}, \frac{5}{2}, \frac{5}{1}, \frac{2}{6}, \frac{2}{3}, \frac{2}{2}, \frac{2}{1}, \frac{1}{6}, \frac{1}{3}, \frac{1}{2}, \frac{1}{1}, -\frac{10}{6}, -\frac{10}{3}, -\frac{10}{2}, -\frac{10}{1}, -\frac{5}{6}, -\frac{5}{3}, -\frac{5}{2}, -\frac{5}{1}, -\frac{2}{6}, -\frac{2}{3}, -\frac{2}{2}... \) and so on, after simplification.

Key concepts

Leading Coefficient

In the polynomial function given, \(f(x) = -6x^3 - x^2 + x + 10\), the leading coefficient is the number in front of the term with the highest degree. Here, the highest degree term is \(-6x^3\), making the leading coefficient \(-6\).
The leading coefficient helps determine the end behavior of the polynomial. For example, since \(-6\) is negative, as \(x\) goes to infinity, the polynomial will drop towards negative infinity.

Constant Term

The constant term in a polynomial is the term without any variables. In the polynomial \(f(x) = -6x^3 - x^2 + x + 10\), the constant term is \(10\).
The constant term plays an essential role in identifying potential rational zeros, as its factors are used to generate a list of possible rational solutions.

Factors of a Polynomial

To find the potential rational zeros of a polynomial, you need the factors of the constant term and the factors of the leading coefficient. In our example, the constant term is \(10\), and its factors are \(\text{-10, -5, -2, -1, 1, 2, 5, 10}\). The leading coefficient is \(-6\), and its factors are \(-6, -3, -2, -1, 1, 2, 3, 6\).
Factors are numbers that divide evenly into another number. They help form the basis for the Rational Zeros Theorem, allowing the list of potential rational zeros to be created.

Rational Zeros Theorem

The Rational Zeros Theorem is a useful tool in polynomial algebra. This theorem states that if a polynomial has a rational zero, it is a fraction of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient.
Using the given polynomial \(f(x) = -6x^3 - x^2 + x + 10\), we form potential rational zeros by dividing each factor of \(10\) (the constant term) by each factor of \(-6\) (the leading coefficient).
This generates many fractions, which can be simplified to achieve a list of potential rational zeros. Remember, the theorem gives possible zeros but does not guarantee all of them are actual solutions.