Problem 433 - Precalculus

Question

List all possible rational zeros for the functions. $$f(x)=6 x^{4}-10 x^{2}+13 x+1$$

Step-by-Step Solution

Answer

The possible rational zeros are \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6} \).

Step 1: Understanding Rational Zero Theorem

The Rational Zero Theorem states that if a polynomial has a rational zero \( \frac{p}{q} \), then \( p \) (the numerator) must be a factor of the constant term, and \( q \) (the denominator) must be a factor of the leading coefficient.

Step 2: Identify Constant Term and Leading Coefficient

For the function \( f(x) = 6x^4 - 10x^2 + 13x + 1 \), the constant term is 1 and the leading coefficient is 6.

Step 3: List Factors of Constant Term

The factors of the constant term 1 are \( \pm 1 \).

Step 4: List Factors of Leading Coefficient

The factors of the leading coefficient 6 are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Step 5: List Possible Rational Zeros

Combine the factors from Step 3 and Step 4 to find possible rational zeros. The possible rational zeros are \( \frac{\pm 1}{\pm 1}, \frac{\pm 1}{\pm 2}, \frac{\pm 1}{\pm 3}, \frac{\pm 1}{\pm 6} \). Simplifying these, we find the possible rational zeros to be: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6} \).

Key Concepts

Polynomials

Polynomials are expressions composed of variables and coefficients that involve only the operations of addition, subtraction, and multiplication, along with non-negative integer exponents of variables. A polynomial can have one or more terms, and each term is a product of a constant and a power of a variable. The general form is given by

• \( a_n x^n + a_{n-1} x^{n-1} + ext{...} + a_2 x^2 + a_1 x + a_0 \)

In this problem, the polynomial provided is a quartic polynomial, which means it has a degree of four due to the highest exponent, \( x^4 \). Each term in the polynomial contributes to its overall behavior and properties. Recognizing how each component affects the polynomial will help in understanding the Rational Zero Theorem and list possible rational zeros.

Constant Term

The constant term in a polynomial is the term that does not contain any variables, meaning the variable part is essentially raised to the power of zero. It's important because it affects the intercept of the polynomial on the graph's y-axis when evaluating the function at zero. In the polynomial

• \( f(x) = 6x^4 - 10x^2 + 13x + 1 \)

the constant term is 1. Identifying the constant term is crucial in using the Rational Zero Theorem because the theorem states that the possible values for the numerator \( p \) of any rational zero \( \frac{p}{q} \) are factors of this constant term.

The Importance in Finding Rational Zeros

To find the potential rational zeros, you need to list the factors of the constant term. Since the constant term here is 1, its factors are simply \( \pm 1 \). This makes it very straightforward in calculating potential zeros as it simplifies the possible numerator values.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It plays a significant role in determining the end behavior of the polynomial's graph. In our polynomial example

• \( f(x) = 6x^4 - 10x^2 + 13x + 1 \)

the leading coefficient is 6 from the term \( 6x^4 \). Recognizing the leading coefficient is vital as it contributes to the denominator part ``\( q \)`` when applying the Rational Zero Theorem.

The Role in Rational Zero Theorem

According to the theorem, potential rational zeros have denominators that are factors of the leading coefficient. For the leading coefficient of 6, these factors include \( \pm 1, \pm 2, \pm 3, \pm 6 \). Listing these helps in forming the set of possible rational zeros when combined with the factors of the constant term.

Factors

Factors are numbers that can be multiplied together to arrive at another number. In the context of polynomials, determining factors is a key step in the Rational Zero Theorem to find possible rational zeros.

Determining Factors

The step-by-step solution of the original problem demonstrates finding factors of the constant term and the leading coefficient:

• The constant term 1 has factors of \( \pm 1 \).
• The leading coefficient 6 has factors of \( \pm 1, \pm 2, \pm 3, \pm 6 \).

To determine the possible rational zeros, these sets of factors combine as numerators and denominators to create fractions that represent potential rational zeros.

Forming Possible Rational Zeros

By considering combinations of these factors in the form of \( \frac{p}{q} \), potential rational zeros include \( \frac{\pm 1}{\pm 1}, \frac{\pm 1}{\pm 2}, \frac{\pm 1}{\pm 3}, \frac{\pm 1}{\pm 6} \). Simplifying these gives \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6} \). Each value must be verified by substituting back into the original polynomial to confirm if they are indeed zeros. This method is fundamental to algebraic problem-solving and facilitates understanding the relationship between a polynomial's structure and its potential solutions.