Finding Roots Of Cubic Function F(x) = 4x³ - 4x² - 16x + 16 Using Remainder Theorem
Introduction
In this article, we delve into the process of finding all the roots of the cubic function f(x) = 4x³ - 4x² - 16x + 16, given that one factor is (x - 2). We will leverage the Remainder Theorem and polynomial division to systematically identify all the roots. Understanding how to find roots of polynomial functions is crucial in various fields of mathematics, engineering, and computer science. This article provides a step-by-step guide, making it easy for anyone to follow along and grasp the concepts.
Understanding the Remainder Theorem
Before we dive into the specifics of our function, let's briefly recap the Remainder Theorem. The Remainder Theorem states that if a polynomial f(x) is divided by x - a, the remainder is f(a). In simpler terms, if we substitute x = a into the polynomial, the result we get is the remainder of the division. If f(a) = 0, then x - a is a factor of f(x), and a is a root of the function. In our case, we're given that (x - 2) is a factor, which means f(2) = 0. This information is a crucial starting point for finding the other roots of the polynomial. Knowing the Remainder Theorem allows us to simplify complex polynomial equations into more manageable forms. This theorem not only helps in identifying factors but also aids in reducing the degree of the polynomial, making it easier to solve. For instance, if we divide a cubic polynomial by a linear factor, we are left with a quadratic polynomial, which can be solved using various methods like factoring, completing the square, or the quadratic formula. The Remainder Theorem is a cornerstone in polynomial algebra, bridging the gap between polynomial division and the evaluation of polynomial functions. This foundational understanding is vital for tackling problems involving polynomial factorization and root finding.
Step 1: Verify (x - 2) is a Factor
As a first step, let’s verify that (x - 2) is indeed a factor of f(x) = 4x³ - 4x² - 16x + 16. According to the Remainder Theorem, if (x - 2) is a factor, then f(2) should equal zero. We substitute x = 2 into the function:
f(2) = 4(2)³ - 4(2)² - 16(2) + 16 f(2) = 4(8) - 4(4) - 32 + 16 f(2) = 32 - 16 - 32 + 16 f(2) = 0
Since f(2) = 0, we confirm that (x - 2) is a factor of f(x). This verification is not just a formality; it reinforces our understanding of the problem and sets the stage for the next steps. By verifying the given factor, we ensure that our subsequent calculations are based on solid ground. This step also highlights the practical application of the Remainder Theorem, showing how it can be used to quickly check if a given binomial is a factor of a polynomial. Moreover, this process helps in developing a systematic approach to problem-solving, which is invaluable in mathematics. Verifying the initial condition ensures accuracy and provides confidence in the solution process. This preliminary step often saves time and effort in the long run by preventing errors that might arise from incorrect assumptions.
Step 2: Polynomial Division
Now that we have confirmed that (x - 2) is a factor, we proceed to divide f(x) by (x - 2) using polynomial long division. This process will help us find the remaining quadratic factor. Polynomial division involves dividing the given polynomial by a known factor to reduce the polynomial to a lower degree. Here’s how we perform the long division:
4x² + 4x - 8
x - 2 | 4x³ - 4x² - 16x + 16
4x³ - 8x²
------------
4x² - 16x
4x² - 8x
----------
-8x + 16
-8x + 16
--------
0
The quotient we obtain from this division is 4x² + 4x - 8. This means that f(x) can be expressed as:
f(x) = (x - 2)(4x² + 4x - 8)
Polynomial division is a fundamental technique in algebra for breaking down complex polynomials into simpler components. This step is crucial because it reduces the cubic polynomial into a product of a linear factor and a quadratic factor. The quadratic factor is much easier to solve for roots compared to the original cubic polynomial. Understanding polynomial division not only aids in finding roots but also in simplifying algebraic expressions and solving equations. The process involves systematically dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor. The quotient obtained from this division provides valuable information about the factors of the polynomial.
Step 3: Find the Roots of the Quadratic Factor
We now need to find the roots of the quadratic factor, 4x² + 4x - 8. To simplify, we can first factor out the common factor of 4:
4(x² + x - 2)
Now, we focus on finding the roots of the quadratic x² + x - 2. We can factor this quadratic expression:
(x² + x - 2) = (x + 2)(x - 1)
Setting each factor equal to zero gives us the roots:
x + 2 = 0 => x = -2 x - 1 = 0 => x = 1
Thus, the roots of the quadratic factor are x = -2 and x = 1. Finding the roots of a quadratic equation is a critical skill in algebra. Factoring, as demonstrated here, is one of the most efficient methods when the quadratic expression can be easily factored. The process involves breaking down the quadratic expression into two binomial factors. Each factor is then set to zero, and the resulting linear equations are solved to find the roots. In cases where factoring is not straightforward, other methods such as completing the square or using the quadratic formula can be employed. The quadratic formula, in particular, is a universally applicable method for finding the roots of any quadratic equation, regardless of its factorability. Understanding these techniques is essential for solving a wide range of mathematical problems, from simple equations to more complex applications in calculus and other advanced topics. The ability to find roots of quadratic equations is a fundamental building block in mathematical education.
Step 4: Identify All Roots
Combining the root from the given factor (x - 2) and the roots from the quadratic factor, we find all the roots of f(x). From (x - 2), we have x = 2. From the quadratic factor, we have x = -2 and x = 1. Therefore, the roots of f(x) = 4x³ - 4x² - 16x + 16 are x = 2, x = -2, and x = 1. Identifying all roots of a polynomial function is a critical step in understanding its behavior. The roots, also known as zeros, are the points where the function intersects the x-axis. These values provide significant information about the function's graph and its properties. In the case of a cubic function, there can be up to three real roots, as we have found here. The process of finding all roots often involves a combination of techniques, such as factoring, polynomial division, and the use of the Remainder Theorem. Each root corresponds to a factor of the polynomial, and knowing the roots allows us to express the polynomial in its fully factored form. This form is invaluable for further analysis, including sketching the graph of the function and solving related equations and inequalities. The ability to identify all roots is a fundamental skill in algebra and is essential for more advanced mathematical studies.
Conclusion
In summary, we found all the roots of the function f(x) = 4x³ - 4x² - 16x + 16 by using the Remainder Theorem and polynomial division. Given that (x - 2) was a factor, we divided the polynomial by this factor to obtain a quadratic expression. We then found the roots of the quadratic expression by factoring. Combining these roots with the root from the given factor, we determined that the roots of f(x) are x = 2, x = -2, and x = 1. This methodical approach demonstrates how complex problems can be broken down into simpler steps, making them more manageable. The Remainder Theorem and polynomial division are powerful tools in algebra that allow us to solve for roots of polynomial functions efficiently. Understanding these techniques not only helps in finding solutions but also in deepening our understanding of polynomial behavior. The process of finding roots is fundamental in many areas of mathematics and its applications, making it a crucial skill for students and professionals alike. By mastering these methods, we can tackle a wide range of problems involving polynomial equations and functions.
