Polynomial Division Find (f/g)(x) And (f/g)(1) For F(x) And G(x)

Introduction

In this article, we will delve into the fascinating world of polynomial division. Our focus will be on finding the quotient of two polynomials, specifically when $f(x) = 4x^4 - 19x^3 - 10x^2 + 35x - 50$ is divided by $g(x) = x - 5$. We aim to determine $\left(\frac{f}{g}\right)(x)$, which represents the resulting polynomial after the division, and then evaluate this quotient at $x = 1$ to find $\left(\frac{f}{g}\right)(1)$. This exploration will involve the application of polynomial long division, a fundamental technique in algebra, and will provide insights into how polynomial functions interact with each other. Understanding these concepts is crucial for various applications in mathematics, engineering, and computer science, where polynomial functions are frequently used to model real-world phenomena. We will also discuss the importance of the remainder theorem and factor theorem in the context of polynomial division. Throughout this discussion, we will emphasize the step-by-step process, ensuring that the underlying principles are clear and accessible. This journey into polynomial division will not only enhance your algebraic skills but also deepen your appreciation for the elegance and power of mathematical tools.

Polynomial Long Division: Finding (f/g)(x)

To find $\left(\frac{f}{g}\right)(x)$, where $f(x) = 4x^4 - 19x^3 - 10x^2 + 35x - 50$ and $g(x) = x - 5$, we will employ the method of polynomial long division. This method systematically divides one polynomial by another, similar to the long division process used for integers. The key is to focus on the leading terms at each step and eliminate them through subtraction. Let's break down the process step by step. First, we set up the long division as follows:

 x - 5 | 4x^4 - 19x^3 - 10x^2 + 35x - 50

Next, we divide the leading term of $f(x)$, which is $4x^4$, by the leading term of $g(x)$, which is $x$. This gives us $4x^3$. We write this above the division symbol and multiply $4x^3$ by $g(x) = x - 5$, which results in $4x^4 - 20x^3$. We then subtract this from the original polynomial:

 4x^3
 x - 5 | 4x^4 - 19x^3 - 10x^2 + 35x - 50
 -(4x^4 - 20x^3)
 ------------------
 x^3 - 10x^2

Now, we bring down the next term, $-10x^2$, and repeat the process. We divide the leading term of the new polynomial, $x^3$, by the leading term of $g(x)$, which is $x$. This gives us $x^2$. We add this to the quotient above and multiply $x^2$ by $g(x) = x - 5$, resulting in $x^3 - 5x^2$. Subtracting this from the current polynomial gives:

 4x^3 + x^2
 x - 5 | 4x^4 - 19x^3 - 10x^2 + 35x - 50
 -(4x^4 - 20x^3)
 ------------------
 x^3 - 10x^2
 -(x^3 - 5x^2)
 ------------------
 -5x^2 + 35x

We bring down the next term, $35x$, and continue. Dividing $-5x^2$ by $x$ gives $-5x$. We add this to the quotient and multiply $-5x$ by $g(x) = x - 5$, which gives $-5x^2 + 25x$. Subtracting this yields:

 4x^3 + x^2 - 5x
 x - 5 | 4x^4 - 19x^3 - 10x^2 + 35x - 50
 -(4x^4 - 20x^3)
 ------------------
 x^3 - 10x^2
 -(x^3 - 5x^2)
 ------------------
 -5x^2 + 35x
 -(-5x^2 + 25x)
 ------------------
 10x - 50

Finally, we bring down the last term, $-50$. Dividing $10x$ by $x$ gives $10$. We add this to the quotient and multiply $10$ by $g(x) = x - 5$, resulting in $10x - 50$. Subtracting this gives a remainder of $0$:

 4x^3 + x^2 - 5x + 10
 x - 5 | 4x^4 - 19x^3 - 10x^2 + 35x - 50
 -(4x^4 - 20x^3)
 ------------------
 x^3 - 10x^2
 -(x^3 - 5x^2)
 ------------------
 -5x^2 + 35x
 -(-5x^2 + 25x)
 ------------------
 10x - 50
 -(10x - 50)
 ------------------
 0

Thus, $\left(\frac{f}{g}\right)(x) = 4x^3 + x^2 - 5x + 10$.

Evaluating (f/g)(1)

Now that we have found $\left(\frac{f}{g}\right)(x) = 4x^3 + x^2 - 5x + 10$, we can evaluate it at $x = 1$ to find $\left(\frac{f}{g}\right)(1)$. This involves substituting $1$ for $x$ in the expression. So, we have:

$\left(\frac{f}{g}\right)(1) = 4(1)^3 + (1)^2 - 5(1) + 10$

Simplifying this expression, we get:

$\left(\frac{f}{g}\right)(1) = 4(1) + 1 - 5 + 10$ $\left(\frac{f}{g}\right)(1) = 4 + 1 - 5 + 10$ $\left(\frac{f}{g}\right)(1) = 10$

Therefore, $\left(\frac{f}{g}\right)(1) = 10$. This result provides a specific value of the quotient polynomial at $x=1$, illustrating the application of polynomial division and evaluation in determining function values.

Remainder and Factor Theorems

In the process of polynomial division, the Remainder Theorem and the Factor Theorem play crucial roles. These theorems provide valuable insights into the relationship between polynomial division, roots, and factors. The Remainder Theorem states that if a polynomial $f(x)$ is divided by $x - c$, the remainder is $f(c)$. In our case, we divided $f(x)$ by $g(x) = x - 5$. If we evaluate $f(5)$, we should get the same remainder as we did in our long division, which was 0. Let's verify this:

$f(5) = 4(5)^4 - 19(5)^3 - 10(5)^2 + 35(5) - 50$ $f(5) = 4(625) - 19(125) - 10(25) + 175 - 50$ $f(5) = 2500 - 2375 - 250 + 175 - 50$ $f(5) = 0$

As expected, the remainder is 0, which confirms the Remainder Theorem. The Factor Theorem is a special case of the Remainder Theorem. It states that $x - c$ is a factor of $f(x)$ if and only if $f(c) = 0$. Since $f(5) = 0$, we can conclude that $x - 5$ is indeed a factor of $f(x)$, which we already knew since the remainder was 0 in our long division. These theorems are not just theoretical constructs; they are powerful tools for simplifying polynomial expressions, finding roots, and understanding the structure of polynomial functions. They allow us to quickly determine factors and remainders without performing the full long division in some cases, significantly streamlining the problem-solving process. By understanding and applying these theorems, we can gain a deeper appreciation for the interconnectedness of different concepts in algebra.

Importance of Polynomial Division

Polynomial division is a fundamental operation in algebra with a wide range of applications in mathematics, engineering, and computer science. Understanding polynomial division is crucial for simplifying complex expressions, solving equations, and analyzing functions. In mathematics, polynomial division is used to factor polynomials, find roots, and determine the behavior of rational functions. Factoring polynomials, in particular, is a cornerstone of solving polynomial equations, as it allows us to break down a complex equation into simpler, more manageable parts. The roots of a polynomial, which are the values of $x$ that make the polynomial equal to zero, can often be found by identifying factors through polynomial division. This is particularly important in applications where we need to find the points at which a function crosses the x-axis. In engineering, polynomial division is used in control systems, signal processing, and circuit analysis. For example, in control systems, the transfer function of a system, which describes the relationship between the input and output, is often a rational function (a ratio of two polynomials). Polynomial division can be used to simplify this transfer function, making it easier to analyze the system's behavior. In signal processing, polynomial division is used in filter design and analysis. Filters are used to remove unwanted noise or signals from a signal, and their behavior can be described using rational functions. In computer science, polynomial division is used in cryptography, coding theory, and computer graphics. For instance, in cryptography, polynomials are used to create encryption algorithms, and polynomial division is used to decode messages. In coding theory, polynomials are used to create error-correcting codes, which are used to detect and correct errors in data transmission. In computer graphics, polynomials are used to represent curves and surfaces, and polynomial division is used in rendering algorithms. The ability to perform polynomial division efficiently and accurately is a critical skill for anyone working in these fields. It allows for the simplification of complex problems and provides insights into the underlying structure of the systems being analyzed. Furthermore, the concepts underlying polynomial division, such as the Remainder Theorem and Factor Theorem, provide a deeper understanding of the behavior of polynomial functions, which is invaluable in a wide range of applications.

Conclusion

In conclusion, we have successfully found $\left(\frac{f}{g}\right)(x) = 4x^3 + x^2 - 5x + 10$ using polynomial long division and evaluated it at $x = 1$ to get $\left(\frac{f}{g}\right)(1) = 10$. This process demonstrates the power and utility of polynomial division in simplifying expressions and evaluating functions. We also explored the Remainder and Factor Theorems, which provide valuable insights into the relationship between polynomial division, roots, and factors. These theorems not only simplify the process of finding remainders and factors but also deepen our understanding of polynomial functions. The significance of polynomial division extends far beyond the classroom, playing a crucial role in various fields such as engineering, computer science, and advanced mathematics. From designing control systems to creating encryption algorithms, the principles of polynomial division underpin many real-world applications. This underscores the importance of mastering this fundamental algebraic technique. By understanding the step-by-step process of polynomial long division and the theoretical underpinnings provided by the Remainder and Factor Theorems, we can tackle complex problems involving polynomial functions with confidence and precision. The journey through polynomial division not only enhances our mathematical skills but also provides a valuable toolset for solving a wide array of problems in various disciplines. As we continue to explore the world of mathematics, the concepts and techniques learned here will serve as a solid foundation for further advancements.