Problem 6 - Algebra and Trigonometry

Question

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=3 x^{4}-11 x^{3}-3 x^{2}-6 x+8 $$

Step-by-Step Solution

Answer

The possible rational zeros for the given function are \(\pm\)1, \(\pm\)2, \(\pm\)4, \(\pm\)8, \(\pm\)1/3, \(\pm\)2/3, \(\pm\)4/3, \(\pm\)8/3.

Step 1: Finding Factors

Find the factors of the constant term, 8, and the leading coefficient, 3. The factors of 8 are ±1, ±2, ±4, and ±8. The factors of 3 are ±1 and ±3.

Step 2: Listing the Possible Rational Zeros

List all possible rational zeros by dividing each factor of the constant term by each factor of the leading coefficient. The possible rational zeros are \(\pm\)1, \(\pm\)2, \(\pm\)4, \(\pm\)8, \(\pm\)1/3, \(\pm\)2/3, \(\pm\)4/3, \(\pm\)8/3.

Step 3: Simplifying Fractions

Simplify the fractions to get the final list of possible rational zeros. The possible simplified rational zeros for the given function are \(\pm\)1, \(\pm\)2, \(\pm\)4, \(\pm\)8, \(\pm\)1/3, \(\pm\)2/3, \(\pm\)4/3, \(\pm\)8/3.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication.
Each term in a polynomial is represented as a power of a variable with a non-negative integer exponent. For example, a polynomial function like \( f(x) = 3x^4 - 11x^3 - 3x^2 - 6x + 8 \) is made up of five terms.
The highest power or exponent, which is 4 in this case, indicates the degree of the polynomial. Understanding the structure of polynomial functions helps in identifying their behavior, such as how they curve and at which points they can cross the x-axis.

Rational Zeros

Rational zeros of a polynomial refer to the values of \( x \) for which the polynomial equals zero, and these values are expressed as fractions \( \frac{p}{q} \) where both \( p \) and \( q \) are integers.
The Rational Zero Theorem provides a strategic way to determine these zeros by considering only those fractions formed by dividing the factors of the constant term by the factors of the leading coefficient.
For the function \( f(x) = 3x^4 - 11x^3 - 3x^2 - 6x + 8 \), the potential rational zeros include values such as \( \pm 1, \pm 2, \pm \frac{1}{3} \), and so on. It is a systematic method for narrowing down which fractions could possibly be zeros, simplifying the trial and error process of finding actual zeros.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a critical role in determining the end behavior of the polynomial's graph.
In the polynomial \( f(x) = 3x^4 - 11x^3 - 3x^2 - 6x + 8 \), the leading coefficient is 3, which is associated with the term \( 3x^4 \).

• The sign of the leading coefficient affects whether the ends of the graph point upward or downward in its extreme limits.
• It also helps us list possible rational zeros by being part of the denominator when creating the fractions for testing possible zeros.

By examining the leading coefficient, we can make predictions about the overall behavior of the polynomial.

Constant Term

The constant term in a polynomial function is the term that does not contain any variable. It is crucial for determining possible rational zeros.
In our polynomial \( f(x) = 3x^4 - 11x^3 - 3x^2 - 6x + 8 \), the constant term is 8.

• The factors of the constant term are candidates for the numerators of potential rational zeros, as suggested by the Rational Zero Theorem.
• For example, using the factors of 8 (\( \pm 1, \pm 2, \pm 4, \pm 8 \)), when paired with the factors of the leading coefficient, they form the possible rational zeros.

Understanding the constant term's role is essential in efficiently testing values to find rational zeros.