The Rational Root Theorem is a powerful tool in algebra that helps us identify potential rational roots of a polynomial equation. In this article, we will delve into the theorem and apply it to the polynomial f(x) = 66x⁴ - 2x³ + 11x² + 35. We aim to determine which statement accurately describes the nature of the rational roots of this polynomial.
Understanding the Rational Root Theorem
The Rational Root Theorem provides a method for finding possible rational roots of a polynomial equation with integer coefficients. It states that if a polynomial f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ has integer coefficients, then any rational root of f(x), when expressed in its simplest form as p/q, must satisfy the following conditions:
- p must be a factor of the constant term a₀.
- q must be a factor of the leading coefficient aₙ.
In essence, the theorem tells us that any rational root of the polynomial must be a fraction where the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient. This significantly narrows down the possibilities when searching for rational roots.
To fully grasp the theorem, let's break down its components and implications:
- Polynomial with Integer Coefficients: The Rational Root Theorem applies specifically to polynomials where all the coefficients (aₙ, aₙ₋₁, ..., a₁, a₀) are integers. This is a crucial requirement, as the theorem's logic relies on the properties of integer division and factorization.
- Rational Root: A rational root of a polynomial f(x) is a rational number x = p/q (where p and q are integers and q ≠ 0) such that f(p/q) = 0. In other words, it's a rational number that, when substituted into the polynomial, makes the polynomial equal to zero. These roots correspond to the x-intercepts of the polynomial's graph.
- Factors of the Constant Term (a₀): The constant term, a₀, is the term in the polynomial that does not have a variable (x) attached to it. The factors of a₀ are all the integers that divide evenly into a₀. For example, if a₀ = 6, its factors are ±1, ±2, ±3, and ±6. These factors become the potential numerators (p) of our rational roots.
- Factors of the Leading Coefficient (aₙ): The leading coefficient, aₙ, is the coefficient of the term with the highest power of x (xⁿ). The factors of aₙ are all the integers that divide evenly into aₙ. For instance, if aₙ = 4, its factors are ±1, ±2, and ±4. These factors become the potential denominators (q) of our rational roots.
- Possible Rational Roots (p/q): The Rational Root Theorem states that any rational root of the polynomial must be in the form of p/q, where p is a factor of a₀ and q is a factor of aₙ. By listing out all possible combinations of p and q, we generate a list of potential rational roots. It's important to note that these are just potential roots; not all of them will necessarily be actual roots of the polynomial. We need to test these potential roots to see if they satisfy the equation f(p/q) = 0.
Applying the Theorem to f(x) = 66x⁴ - 2x³ + 11x² + 35
Now, let's apply the Rational Root Theorem to the given polynomial, f(x) = 66x⁴ - 2x³ + 11x² + 35. First, we identify the key coefficients:
- Leading coefficient (aₙ): 66
- Constant term (a₀): 35
Next, we list the factors of each:
- Factors of 35: ±1, ±5, ±7, ±35
- Factors of 66: ±1, ±2, ±3, ±6, ±11, ±22, ±33, ±66
According to the Rational Root Theorem, any rational root of f(x) must be of the form p/q, where p is a factor of 35 and q is a factor of 66. Therefore, any rational root will be a fraction formed by dividing a factor of 35 by a factor of 66. This means that the possible rational roots are:
±1/1, ±5/1, ±7/1, ±35/1, ±1/2, ±5/2, ±7/2, ±35/2, ±1/3, ±5/3, ±7/3, ±35/3, ±1/6, ±5/6, ±7/6, ±35/6, ±1/11, ±5/11, ±7/11, ±35/11, ±1/22, ±5/22, ±7/22, ±35/22, ±1/33, ±5/33, ±7/33, ±35/33, ±1/66, ±5/66, ±7/66, ±35/66
This list represents all the potential rational roots of the polynomial. To determine the actual rational roots, we would need to test each of these values by substituting them into the polynomial f(x) and checking if the result is zero. If f(p/q) = 0, then p/q is a rational root of the polynomial.
Analyzing the Given Statements
Based on our application of the Rational Root Theorem, we can now evaluate the given statements about the polynomial f(x) = 66x⁴ - 2x³ + 11x² + 35.
The statement "Any rational root of f(x) is a factor of 35 divided by a factor of 66" is true. This aligns perfectly with the conclusion we reached by applying the Rational Root Theorem. The theorem explicitly states that any rational root must be in the form p/q, where p is a factor of the constant term (35) and q is a factor of the leading coefficient (66).
The other options are incorrect because they misrepresent the Rational Root Theorem. For example, a rational root is not necessarily a multiple of 35 or 66, but rather a fraction formed by their factors.
Conclusion
In conclusion, the Rational Root Theorem is a valuable tool for identifying potential rational roots of polynomial equations. By understanding and applying the theorem, we can effectively narrow down the possibilities and find the rational roots of a polynomial. For the polynomial f(x) = 66x⁴ - 2x³ + 11x² + 35, the correct statement, according to the Rational Root Theorem, is that any rational root is a factor of 35 divided by a factor of 66.
This exploration highlights the importance of mastering fundamental theorems in algebra to solve complex problems efficiently. The Rational Root Theorem provides a systematic approach to finding rational roots, making it an indispensable tool for students and professionals alike.
