Finding The Roots Of F(x) = 2x^2 + 2x - 24

In mathematics, determining the roots of a polynomial function is a fundamental task. The roots of a function, also known as zeros, are the values of x for which the function f(x) equals zero. These roots represent the points where the graph of the function intersects the x-axis. For a quadratic function like f(x) = 2x^2 + 2x - 24, finding the roots involves identifying the values of x that satisfy the equation 2x^2 + 2x - 24 = 0. The Rational Root Theorem provides a systematic way to narrow down the possible rational roots of a polynomial equation, which are potential candidates for the actual roots. This theorem is particularly useful when dealing with polynomials that may not be easily factorable or when a quick determination of rational roots is desired. The theorem states that if a polynomial equation with integer coefficients has rational roots, those roots must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. By applying this theorem, we can generate a list of potential rational roots and then test each one to see if it satisfies the equation. In the given problem, the Rational Root Theorem has already provided us with a list of potential roots: -4, -3, 2, 3, and 4. Our task is to determine which of these are the actual roots of the function f(x) = 2x^2 + 2x - 24. This involves substituting each potential root into the function and checking if the result is zero. If f(a) = 0, then a is a root of the function. This process of verification is crucial in solving polynomial equations, especially in contexts where the solutions are not immediately obvious or easily derived through simple algebraic manipulation. Understanding and applying the Rational Root Theorem, along with the method of verifying potential roots, is a cornerstone of polynomial algebra and is essential for more advanced mathematical concepts.

Understanding the Rational Root Theorem

The Rational Root Theorem is a powerful tool in algebra that helps us find potential rational roots of polynomial equations. It's like having a roadmap that guides us to the possible solutions without having to guess randomly. This theorem is particularly useful when dealing with polynomials with integer coefficients, as it provides a systematic way to narrow down the candidates for roots. The essence of the Rational Root Theorem lies in its ability to link the factors of the constant term and the leading coefficient of a polynomial to its potential rational roots. Specifically, the theorem states that if a polynomial equation with integer coefficients has a rational root, that root must be expressible in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. To illustrate, consider a polynomial equation like ax^n + bx^(n-1) + ... + c = 0, where a, b, and c are integers. The Rational Root Theorem tells us that any rational root of this equation must be a fraction where the numerator is a factor of c and the denominator is a factor of a. This provides a structured way to generate a list of potential rational roots. For instance, if the constant term c has factors ±1, ±2, and ±3, and the leading coefficient a has factors ±1 and ±2, then the potential rational roots would be ±1, ±1/2, ±2, ±3, ±3/2. Once we have this list, we can test each potential root by substituting it into the polynomial equation. If the result is zero, then that value is a root of the polynomial. The Rational Root Theorem doesn't guarantee that any of the potential roots are actual roots, but it significantly reduces the number of values we need to check, making the process of solving polynomial equations more efficient. This theorem is a fundamental concept in algebra and is a crucial step in solving higher-degree polynomial equations where factoring or other methods may not be straightforward.

Applying the Rational Root Theorem to f(x) = 2x^2 + 2x - 24

To effectively apply the Rational Root Theorem to the quadratic function f(x) = 2x^2 + 2x - 24, we first need to identify the key components of the polynomial: the leading coefficient and the constant term. In this case, the leading coefficient is 2 (the coefficient of the x^2 term), and the constant term is -24. According to the Rational Root Theorem, any rational root of this function must be of the form p/q, where p is a factor of the constant term (-24) and q is a factor of the leading coefficient (2). Let's list the factors of -24: ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24. These are all the possible values for p. Next, we list the factors of 2: ±1 and ±2. These are the possible values for q. Now, we form all possible fractions p/q by dividing each factor of -24 by each factor of 2. This gives us the following potential rational roots: ±1, ±1/2, ±2, ±3, ±3/2, ±4, ±6, ±8, ±12, and ±24. This list represents all the potential rational roots of the function f(x) = 2x^2 + 2x - 24. However, the problem statement provides us with a pre-selected subset of potential roots: -4, -3, 2, 3, and 4. This simplifies our task, as we only need to test these values to determine which ones are actual roots of the function. Testing these values involves substituting each one into the function and checking if the result is zero. If f(x) = 0 for a particular value of x, then that value is a root of the function. This process of testing potential roots is a direct application of the Rational Root Theorem and is a critical step in finding the solutions to polynomial equations. By systematically applying the theorem and verifying the potential roots, we can efficiently determine the actual roots of the function.

Verifying the Potential Roots

Now that we have the potential roots (-4, -3, 2, 3, and 4), we need to verify which of them are actual roots of the function f(x) = 2x^2 + 2x - 24. This involves substituting each potential root into the function and checking if the result is zero. Let's start with -4:

f(-4) = 2(-4)^2 + 2(-4) - 24 = 2(16) - 8 - 24 = 32 - 8 - 24 = 0

Since f(-4) = 0, -4 is a root of the function.

Next, let's test -3:

f(-3) = 2(-3)^2 + 2(-3) - 24 = 2(9) - 6 - 24 = 18 - 6 - 24 = -12

Since f(-3) ≠ 0, -3 is not a root of the function.

Now, let's test 2:

f(2) = 2(2)^2 + 2(2) - 24 = 2(4) + 4 - 24 = 8 + 4 - 24 = -12

Since f(2) ≠ 0, 2 is not a root of the function.

Let's test 3:

f(3) = 2(3)^2 + 2(3) - 24 = 2(9) + 6 - 24 = 18 + 6 - 24 = 0

Since f(3) = 0, 3 is a root of the function.

Finally, let's test 4:

f(4) = 2(4)^2 + 2(4) - 24 = 2(16) + 8 - 24 = 32 + 8 - 24 = 16

Since f(4) ≠ 0, 4 is not a root of the function.

By substituting each potential root into the function, we have determined that -4 and 3 are the actual roots of f(x) = 2x^2 + 2x - 24. This process of verification is crucial in confirming the roots and ensuring the accuracy of the solution. It demonstrates a practical application of the Rational Root Theorem and reinforces the understanding of how roots relate to the function's values.

Conclusion: The Actual Roots of f(x)

In conclusion, by applying the Rational Root Theorem and verifying the potential roots, we have successfully identified the actual roots of the quadratic function f(x) = 2x^2 + 2x - 24. The given potential roots were -4, -3, 2, 3, and 4. Through the process of substitution and evaluation, we found that f(-4) = 0 and f(3) = 0, while f(-3) ≠ 0, f(2) ≠ 0, and f(4) ≠ 0. Therefore, the actual roots of the function are -4 and 3. These roots are the x-values where the graph of the function intersects the x-axis, representing the solutions to the equation 2x^2 + 2x - 24 = 0. The Rational Root Theorem provided us with a focused set of potential roots to test, which streamlined the process of finding the solutions. This method is particularly valuable for higher-degree polynomials where direct factoring or other solution techniques may not be readily apparent. The ability to determine the roots of a function is a fundamental skill in algebra and calculus, with applications in various fields, including physics, engineering, and economics. Understanding and applying theorems like the Rational Root Theorem enhances problem-solving capabilities and provides a systematic approach to tackling complex mathematical challenges. The process of verifying potential roots through substitution is a critical step in ensuring the accuracy of the solutions and reinforcing the relationship between the roots and the function's behavior. Ultimately, the identification of the roots allows for a more complete understanding of the function and its properties.

For further information on the Rational Root Theorem and polynomial functions, you can visit Khan Academy's Algebra I section.