Problem 58 - College Algebra

Question

For the following exercises, list all possible rational zeros for the functions. \(f(x)=3 x^{3}+5 x^{2}-5 x+4\)

Step-by-Step Solution

Answer

Possible rational zeros: \(\pm 1, \pm 2, \pm 4, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}\).

Step 1: Finding possible rational zeros (Step 1)

First, identify the coefficients of the polynomial function \(f(x) = 3x^3 + 5x^2 - 5x + 4\). The constant term is \(a_0 = 4\) and the leading coefficient is \(a_n = 3\). According to the Rational Root Theorem, any rational zero, \(\frac{p}{q}\), is such that \(p\) is a factor of the constant term \(4\) and \(q\) is a factor of the leading coefficient \(3\).

Step 2: List factors of constant term (Step 2)

List all factors of the constant term \(4\). The factors of \(4\) are \(\pm 1, \pm 2, \pm 4\).

Step 3: List factors of leading coefficient (Step 3)

List all factors of the leading coefficient \(3\). The factors of \(3\) are \(\pm 1, \pm 3\).

Step 4: Determine all possible rational zeros (Step 4)

Using the Rational Root Theorem, construct all possible rational zeros by creating fractions \(\frac{p}{q}\) where \(p\) is each factor of 4 and \(q\) is each factor of 3. This gives: \(\pm 1, \pm 2, \pm 4, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}\).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression consisting of variables raised to various powers and multiplied by coefficients. These expressions are combined using addition, subtraction, and sometimes multiplication. For example, a polynomial function might look like this:

• \(f(x) = 3x^3 + 5x^2 - 5x + 4\)

Each term in the polynomial includes a coefficient (like 3, 5, -5, or 4) and a variable (like \(x\)) raised to a power. Here, the powers of \(x\) are 3, 2, 1, and 0.
Polynomials are useful in various fields, such as engineering, physics, and economics, for modeling complex situations. Understanding polynomials helps in solving equations and understanding functions more deeply.

Constant Term

The constant term in a polynomial is the term that does not contain any variables. It can be seen as the 'starting point' of the function when the input value is zero. In the polynomial function

• \(f(x) = 3x^3 + 5x^2 - 5x + 4\),

the constant term is 4.
This term is important when using the Rational Root Theorem because the factors of the constant term help in finding potential rational zeros of the polynomial. By breaking down the constant term into its factors, we begin to unlock the roots of the polynomial which may have significant implications in problem-solving.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree (the term with the largest power) in a polynomial. In

• \(f(x) = 3x^3 + 5x^2 - 5x + 4\),

the highest degree term is \(3x^3\), so the leading coefficient is 3.
The leading coefficient is vital in the Rational Root Theorem process, as it determines the possible values of the denominator in the fractions representing potential rational zeros. Recognizing factors of the leading coefficient lets us consider various options for the rational zeros which are

• \(rac{p}{q}\)

where

• \(p\) is a factor of the constant term,
• and \(q\) is a factor of the leading coefficient.

Possible Rational Zeros

The possible rational zeros of a polynomial are potential solutions to the equation, candidates for which make the polynomial equal to zero. To find them, the Rational Root Theorem is employed. This theorem claims that if a polynomial has a rational zero \(\frac{p}{q}\), then:

• \(p\) is a factor of the constant term.
• \(q\) is a factor of the leading coefficient.

For the function

• \(f(x) = 3x^3 + 5x^2 - 5x + 4\),

we first find all factors of the constant term (4), which are

• \(\pm 1, \pm 2, \pm 4\)

and all factors of the leading coefficient (3), which are

• \(\pm 1, \pm 3\).

By forming combinations

• (\(\frac{p}{q}\)),

we obtain potential rational zeros:

• \(\pm 1, \pm 2, \pm 4, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}\).

These represent all rational numbers which could potentially be roots of the given polynomial function.