Problem 48 - Precalculus with Calculus Previews

Question

If \(k\) is a prime number (a positive integer greater than 1 whose only positive integer factors are itself and 1) such that \(k>2,\) then what are the possible rational zeros of \(f(x)=6 x^{4}-9 x^{2}+k\) ?

Step-by-Step Solution

Answer

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

Step 1: Understand the Rational Root Theorem

The Rational Root Theorem states that if a polynomial has any rational roots (or zeros), then they are of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term \( a_0 \), and \( q \) is a factor of the leading coefficient \( a_n \). For our polynomial \( f(x) = 6x^4 - 9x^2 + k \), we need to identify these coefficients.

Step 2: Identify the Coefficients

The polynomial is \( f(x) = 6x^4 - 9x^2 + k \). The leading coefficient \( a_n \) is 6, and the constant term \( a_0 \) is \( k \). Hence, the possible rational roots can be expressed as \( \frac{p}{q} \) where \( p \) divides \( k \) and \( q \) divides 6.

Step 3: List the Possible Values of \( p \)

Since \( k \) is a prime number, its only positive divisors are 1 and \( k \) itself. Thus, \( p \), being a divisor of \( k \), can be \( \pm 1 \) or \( \pm k \).

Step 4: List the Possible Values of \( q \)

The number 6 can be divided by \( \pm 1, \pm 2, \pm 3, \) and \( \pm 6 \). Hence, these are the possible values for \( q \).

Step 5: Formulate Possible Rational Zeros

Using the Rational Root Theorem, the possible rational zeros of \( f(x) \) are all the combinations of \( \frac{p}{q} \), i.e., \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm k, \pm \frac{k}{2}, \pm \frac{k}{3}, \pm \frac{k}{6} \).

Step 6: Recognize Constraints of Zeros

These possible rational zeros depend on whether or not they lead to actual zeros of the polynomial. This set determines candidates whose validity needs to be evaluated by substitution or further analysis if required.

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables, coefficients, and exponents. These expressions can have one or more terms, which are separated by plus or minus signs. A polynomial's degree is the highest exponent of its variables. For example, the polynomial \( f(x) = 6x^4 - 9x^2 + k \) is a fourth-degree polynomial because the highest power of \( x \) is 4. Each coefficient, such as 6 and -9, multiplies a specific power of \( x \), while \( k \) serves as the constant term or the standalone number.Polynomials are fundamental in algebra and are used in equations, functions, and modeling diverse real-world scenarios, such as physics and economics. Understanding polynomials involves recognizing the structure of terms and identifying the role of each part, including the leading coefficient, which dictates the polynomial's highest power, and the constant term, which stands alone without a variable.

Prime Numbers

Prime numbers are special numbers greater than 1 that have no divisors other than 1 and themselves. For instance, the numbers 2, 3, 5, 7, and so forth are prime numbers. These numbers are crucial in number theory because they are the building blocks of the integers, meaning every integer greater than 1 is either a prime or can be factorized into a unique combination of prime numbers.In the context of the problem, \( k \) is a prime number greater than 2. This property of primes ensures that the divisors of \( k \) are very limited, being only \( 1 \) and \( k \). This simplicity simplifies the search for potential rational zeros, as the possible divisors of the polynomial's constant term \( k \) involve only these numbers.

Rational Zeros

Rational zeros of a polynomial refer to any values of \( x \) that satisfy the equation \( f(x) = 0 \), where these roots are rational numbers. The Rational Root Theorem helps in identifying these zeros by stating that any potential rational zero \( \frac{p}{q} \) is formed from the factors of the constant term and the leading coefficient. For the polynomial \( f(x) = 6x^4 - 9x^2 + k \), the constant term is \( k \) and the leading coefficient is 6. To find the rational zeros, we list the factors of each:

• Factors of the constant term \( k \) are \( \pm 1 \) and \( \pm k \), since \( k \) is prime.
• Factors of 6 are \( \pm 1, \pm 2, \pm 3, \) and \( \pm 6 \).

The rational zeros would be combinations of these divisors forming fractions \( \frac{p}{q} \) like \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm k, \pm \frac{k}{2}, \pm \frac{k}{3}, \pm \frac{k}{6} \). These form a complete list of potential candidates to be assessed through substitution to determine actual zeros of the polynomial.