Factoring Cubic Functions Using The Remainder Theorem Find All Factors Of F(x)=x^3-4x^2-20x+48
In this article, we will delve into the process of finding all the factors of the cubic function , given that one of its roots is . We will be employing the Remainder Theorem and polynomial division to achieve this. This exploration is crucial for understanding polynomial factorization and its applications in various mathematical contexts. Polynomial factorization is a fundamental concept in algebra, with wide-ranging applications in solving equations, simplifying expressions, and analyzing functions. The Remainder Theorem provides a powerful tool for identifying factors of polynomials, while polynomial division allows us to systematically break down complex expressions into simpler ones. By mastering these techniques, we can gain a deeper understanding of the behavior of polynomial functions and their relationships to their roots and factors.
Understanding the Remainder Theorem
The Remainder Theorem is a cornerstone of polynomial algebra. It states that if we divide a polynomial f(x) by (x - c), then the remainder is f(c). This theorem is incredibly useful for determining if a given value is a root of a polynomial. In simpler terms, if substituting x = c into the polynomial results in f(c) = 0, then (x - c) is a factor of f(x). This is because a remainder of zero indicates that the division is exact, meaning that (x - c) divides f(x) evenly. This concept is fundamental to factoring polynomials and finding their roots. The Remainder Theorem provides a direct link between the roots of a polynomial and its factors, allowing us to identify factors by simply evaluating the polynomial at specific values. This eliminates the need for trial-and-error division, making the process of factorization more efficient. Furthermore, the Remainder Theorem is closely related to the Factor Theorem, which states that (x - c) is a factor of f(x) if and only if f(c) = 0. Together, these theorems form the basis for many techniques used in polynomial algebra.
In our case, we are given that x = 6 is a root of the function f(x) = x³ - 4x² - 20x + 48. According to the Remainder Theorem, this means that f(6) should equal zero. Let's verify this by substituting x = 6 into the function:
f(6) = (6)³ - 4(6)² - 20(6) + 48 f(6) = 216 - 144 - 120 + 48 f(6) = 0
As expected, f(6) = 0, which confirms that (x - 6) is indeed a factor of f(x). This verification step is crucial to ensure that we are on the right track. By confirming that f(6) = 0, we establish a solid foundation for the subsequent steps in the factorization process. This step also highlights the power of the Remainder Theorem in quickly identifying factors of polynomials. By simply evaluating the polynomial at a given value, we can determine whether the corresponding linear expression is a factor. This is a significant advantage over traditional methods of factorization, which can be more time-consuming and cumbersome.
Polynomial Division: Unveiling the Quadratic Factor
Now that we know (x - 6) is a factor, we can use polynomial division to find the remaining quadratic factor. Polynomial division is a method for dividing a polynomial by another polynomial of a lower degree. In this case, we will divide f(x) = x³ - 4x² - 20x + 48 by (x - 6). This process will give us a quotient, which will be a quadratic polynomial, and a remainder. Since we know (x - 6) is a factor, the remainder should be zero. Polynomial division is a systematic way to break down a polynomial into its factors. It involves a series of steps that are similar to long division with numbers. The process begins by dividing the leading term of the dividend (the polynomial being divided) by the leading term of the divisor (the polynomial we are dividing by). The result is then multiplied by the divisor, and the product is subtracted 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 process represents the remaining factor of the polynomial.
Here's how the polynomial division works:
x² + 2x - 8
x - 6 | x³ - 4x² - 20x + 48
-(x³ - 6x²)
----------------
2x² - 20x
-(2x² - 12x)
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-8x + 48
-(-8x + 48)
----------
0
The quotient we obtain is x² + 2x - 8. This means that:
x³ - 4x² - 20x + 48 = (x - 6)(x² + 2x - 8)
This step is crucial because it reduces the cubic polynomial to a product of a linear factor and a quadratic factor. The quadratic factor is easier to handle and can be further factored using standard techniques such as factoring by grouping or using the quadratic formula. The process of polynomial division provides a systematic way to find the remaining factors of a polynomial once one factor is known. This is a powerful tool in polynomial algebra, allowing us to break down complex polynomials into simpler components that are easier to analyze and manipulate. By dividing the original polynomial by the known factor, we can obtain a quotient that represents the remaining factors. This quotient can then be further analyzed to find additional factors, if any.
Factoring the Quadratic: The Final Pieces
Now, we need to factor the quadratic expression x² + 2x - 8. We are looking for two numbers that multiply to -8 and add to 2. These numbers are 4 and -2. Therefore, we can factor the quadratic as follows:
x² + 2x - 8 = (x + 4)(x - 2)
This step involves factoring a quadratic expression, which is a fundamental skill in algebra. Factoring a quadratic involves finding two binomial expressions that, when multiplied together, produce the original quadratic. There are several techniques for factoring quadratics, including factoring by grouping, using the quadratic formula, and trial and error. In this case, we used the method of finding two numbers that multiply to the constant term (-8) and add to the coefficient of the linear term (2). These numbers are 4 and -2, which allows us to factor the quadratic as (x + 4)(x - 2). This step is crucial in completing the factorization of the original cubic polynomial. By factoring the quadratic expression, we break it down into two linear factors, which are the final pieces of the puzzle. This allows us to express the cubic polynomial as a product of three linear factors, which provides a complete understanding of its roots and behavior.
Putting It All Together: The Complete Factorization
Combining the factors, we have:
f(x) = (x - 6)(x² + 2x - 8) f(x) = (x - 6)(x + 4)(x - 2)
Thus, the factors of the function f(x) = x³ - 4x² - 20x + 48 are (x - 6), (x + 4), and (x - 2).
This final step combines all the previous steps to arrive at the complete factorization of the original cubic polynomial. We started by using the Remainder Theorem to identify one factor (x - 6). Then, we used polynomial division to divide the cubic polynomial by this factor, resulting in a quadratic quotient. Finally, we factored the quadratic quotient into two linear factors. By combining these three linear factors, we obtain the complete factorization of the cubic polynomial. This factorization provides valuable information about the roots of the polynomial, which are the values of x that make the polynomial equal to zero. The roots can be easily determined from the factors by setting each factor equal to zero and solving for x. In this case, the roots are x = 6, x = -4, and x = 2. The complete factorization also provides insights into the behavior of the polynomial function, such as its intercepts, turning points, and end behavior. Understanding the factorization of a polynomial is therefore crucial for analyzing and manipulating polynomial functions.
Conclusion
By applying the Remainder Theorem and polynomial division, we successfully factored the cubic function f(x) = x³ - 4x² - 20x + 48. We found that the factors are (x - 6), (x + 4), and (x - 2). This process demonstrates a powerful method for factoring polynomials, which is a fundamental skill in algebra and calculus. Mastering these techniques allows us to solve polynomial equations, simplify expressions, and analyze the behavior of polynomial functions. The Remainder Theorem provides a quick way to identify factors, while polynomial division allows us to systematically break down complex polynomials into simpler ones. By combining these techniques, we can efficiently factor polynomials and gain a deeper understanding of their properties. This knowledge is essential for further studies in mathematics and related fields, where polynomial functions play a crucial role in modeling and solving real-world problems. The ability to factor polynomials is a valuable asset in any mathematical toolkit, enabling us to tackle a wide range of problems with confidence and efficiency.
Therefore, the correct answer is C. (x-2)(x+4)(x-6)
