Problem 2 - College Algebra

Question

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=x^{3}+3 x^{2}-6 x-8 $$

Step-by-Step Solution

Answer

All possible rational zeros for the function \(f(x)=x^{3}+3x^{2}-6x-8\) are -1, 1, -2, 2, -4, 4, -8, 8.

Step 1: Identify coefficients

To start, identify your constant term and leading coefficient from the given polynomial. In this case, your leading coefficient (the coefficient of \(x^3\)) is 1, and your constant term (the term that doesn't include \(x\)) is -8.

Step 2: Factor coefficients

Next, you need to find all factors of 1 and -8. Factors of 1 are only 1 and -1, because 1 is a prime number. Factors of -8 are -1, 1, -2, 2, -4, 4, -8, and 8.

Step 3: List possible rational zeros

After finding all factors, in this case every factor of -8 will be divided by each factor of 1. Because 1 and -1 are the only factors of the leading coefficient (which is 1), for this example, the rational zeros will be the same as factors of -8. These zeros are -1, 1, -2, 2, -4, 4, -8, 8.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables and coefficients that include addition, subtraction, multiplication, and non-negative integer exponents of variables. They can be expressed in the form of
\( f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x^1 + a_0 \)
where
\( a_n, a_{n-1}, \ldots, a_1, a_0 \) are coefficients and
\( n \) is a non-negative integer representing the degree of the polynomial. The highest exponent of the variable
\( x \) indicates the degree of the polynomial.

The degree of a polynomial function provides information about the behavior of the graph of the function, such as the number of turning points and the end behavior. Polynomial functions play a major role in various areas of mathematics and science due to their simplicity and the ease with which they can model complex systems.

Rational Zeros

Rational zeros, also known as rational roots, are possible values of
\( x \) that satisfy
\( f(x) = 0 \) when
\( f(x) \) is a polynomial equation with integer coefficients. The Rational Zero Theorem provides a handy way to find all potential rational zeros of a polynomial function. According to the theorem, if a polynomial
\( f(x) \) has rational zeros, they are of the form
\( \pm \frac{p}{q} \) where
\( p \) is a factor of the constant term and
\( q \) is a factor of the leading coefficient.

When applying the theorem, as in our example exercise, list out the factors of the constant term and those of the leading coefficient. Then, taking each possible combination of
\( \pm \frac{p}{q} \) gives us a list of all possible rational zeros. This list is not a guarantee that those numbers are actual zeros, but a starting point for further investigation, usually through synthetic division or other methods of polynomial evaluation.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This can simplify polynomial expressions and solve polynomial equations, as it can reveal the roots or zeros of the polynomial function. There are various factoring methods including:

• Greatest Common Factor (GCF)
• Grouping
• Difference of squares
• Sum or difference of cubes
• Quadratic form

In the context of finding zeros, after listing all possible rational zeros using the Rational Zero Theorem, one can use these factoring techniques to test which of the potential zeros are actual zeros of the polynomial. Factoring is an essential skill in algebra, as it is required for simplifying expressions, dividing polynomials, and solving polynomial equations.