Problem 58 - College Algebra with Corequisite Support

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

Step 1: Identify the Polynomial Coefficients

The polynomial given is \( f(x) = 3x^3 + 5x^2 - 5x + 4 \). The coefficients of the polynomial are 3 (leading coefficient) and the constant term is 4.

Step 2: Apply the Rational Root Theorem

The Rational Root Theorem states that any rational root of the polynomial equation \(a_0x^n + a_{n-1}x^{n-1} + ... + a_{n-1}x + a_n = 0\) is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_n\) and \(q\) is a factor of the leading coefficient \(a_0\).

Step 3: Find Factors of Constant and Leading Coefficient

List the factors of the constant term (4): \(\pm 1, \pm 2, \pm 4\). List the factors of the leading coefficient (3): \(\pm 1, \pm 3\).

Step 4: Create Possible Rational Zeros

Divide each factor of the constant term by each factor of the leading coefficient to find all possible rational roots. These are: \(\pm 1, \pm 2, \pm 4, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}\).

Key Concepts

Polynomial Coefficients

In any polynomial, coefficients are the numbers placed in front of and multiplying the variable terms. For example, consider the polynomial function \( f(x) = 3x^3 + 5x^2 - 5x + 4 \). Each term like \( 3x^3 \), \( 5x^2 \), and \(-5x\) has a coefficient:

• The coefficient of \( x^3 \) is 3.
• The coefficient of \( x^2 \) is 5.
• The coefficient of \( x \) is -5.
• The constant term is 4, which does not have a variable but acts as the coefficient of \( x^0 \).

Understanding coefficients is essential when applying the Rational Root Theorem, as it involves finding factors of these coefficients. Coefficients influence the behavior and the graph of the polynomial.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. It significantly impacts the properties of the polynomial's graph, especially as \( x \) approaches positive or negative infinity.

For the polynomial \( f(x) = 3x^3 + 5x^2 - 5x + 4 \), the leading term is \( 3x^3 \), making the leading coefficient 3. This coefficient helps determine possible rational roots, as per the Rational Root Theorem.

When listing possible rational edges, you'll use factors of both the leading coefficient and the constant term. For our polynomial, the leading coefficient's factors are \( \pm 1 \) and \( \pm 3 \). These factors help define the denominator in the potential rational root fractions.

Constant Term

The constant term of a polynomial is the term that does not include any variables, a.k.a., it is simply a standalone number. In \( f(x) = 3x^3 + 5x^2 - 5x + 4 \), the constant term is 4. The constant term typically influences the y-intercept of the polynomial's graph.

When finding rational roots using the Rational Root Theorem, the constant term is crucial. It gives the possible numerators for the rational roots.

• For our example, the factors of 4 are \( \pm 1 \), \( \pm 2 \), and \( \pm 4 \).

These factors will be used in combination with factors of the leading coefficient to consider all possible rational zeros.

Possible Rational Zeros

The Rational Root Theorem helps predict possible rational zeros (or roots) of a polynomial equation. A rational zero can be expressed as a fraction \( \frac{p}{q} \) where:

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

To find the possible rational zeros of \( f(x) = 3x^3 + 5x^2 - 5x + 4 \), we use:

• Factors of the constant term (4): \( \pm 1, \pm 2, \pm 4 \).
• Factors of the leading coefficient (3): \( \pm 1, \pm 3 \).

By dividing each factor of the constant term by each factor of the leading coefficient, we obtain the set of possible rational zeros: \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3} \). These potential roots can then be tested or validated within the original polynomial function.