Finding The Quotient Polynomial Dividing (x^4 - 3x^2 + 4x - 3) By (x^2 + X - 3)

Jul 15, 2025 by ADMIN 80 views

Unveiling the Quotient Polynomial Resulting from Dividing (x^4 - 3x^2 + 4x - 3) by (x^2 + x - 3)

Introduction

In this comprehensive exploration, we embark on a mathematical journey to determine the quotient polynomial obtained when dividing the polynomial (x^4 - 3x^2 + 4x - 3) by the polynomial (x^2 + x - 3). Polynomial division, a fundamental concept in algebra, allows us to break down complex expressions into simpler forms, revealing hidden relationships and facilitating further analysis. This article delves into the step-by-step process of polynomial long division, providing a clear and concise explanation of each operation involved. We aim to not only find the quotient but also to illuminate the underlying principles of polynomial division, making this topic accessible to learners of all levels. The concepts covered here serve as a building block for more advanced algebraic manipulations, emphasizing the importance of mastering polynomial division for success in higher mathematics. Through meticulous calculations and detailed explanations, we will demystify the process of polynomial division and equip you with the skills to tackle similar problems with confidence.

Setting Up the Polynomial Long Division

To initiate the process of polynomial long division, we must arrange the polynomials in the correct format. The dividend, which is the polynomial being divided (x^4 - 3x^2 + 4x - 3), is placed inside the long division symbol, while the divisor, the polynomial we are dividing by (x^2 + x - 3), is placed outside. It is crucial to ensure that both polynomials are written in descending order of their exponents, with any missing terms represented by coefficients of zero. This step is essential for maintaining proper alignment and organization during the division process. In our case, we observe that the dividend is missing the x^3 term. Therefore, we rewrite it as (x^4 + 0x^3 - 3x^2 + 4x - 3). This ensures that each power of x has a corresponding term, facilitating the accurate execution of the long division algorithm. Similarly, the divisor (x^2 + x - 3) is already in the correct format. The setup is the foundation upon which we build our solution, and a clear and organized setup minimizes the likelihood of errors in the subsequent steps. The correct arrangement of terms and the inclusion of zero coefficients for missing terms are critical for the success of the polynomial long division.

Step-by-Step Polynomial Long Division

Step 1: Divide the Leading Terms

We commence the long division process by focusing on the leading terms of both the dividend and the divisor. In this case, the leading term of the dividend is x^4, and the leading term of the divisor is x^2. Our first task is to determine what we must multiply the divisor's leading term (x^2) by to obtain the dividend's leading term (x^4). The answer is x^2, since x^2 * x^2 = x^4. This x^2 becomes the first term of our quotient, which we write above the division symbol, aligned with the x^4 term in the dividend. This step sets the foundation for the rest of the division process, ensuring that we systematically reduce the degree of the polynomial being divided. The accurate determination of this first quotient term is paramount, as it dictates the subsequent terms and the final result. We must meticulously ensure that we are targeting the leading terms and performing the correct multiplication to proceed smoothly through the algorithm.

Step 2: Multiply the Quotient Term by the Divisor

Having determined the first term of the quotient as x^2, we now proceed to multiply this term by the entire divisor (x^2 + x - 3). This multiplication yields x^2 * (x^2 + x - 3) = x^4 + x^3 - 3x^2. The resulting polynomial, x^4 + x^3 - 3x^2, is then written below the corresponding terms in the dividend. Proper alignment of terms with the same degree is crucial at this stage, as it ensures accurate subtraction in the subsequent step. This multiplication step is a critical component of the long division process, as it represents the portion of the dividend that is accounted for by the current quotient term. A careful and precise multiplication is necessary to maintain the integrity of the algorithm and avoid errors that could propagate through the rest of the division.

Step 3: Subtract and Bring Down the Next Term

Now we subtract the polynomial (x^4 + x^3 - 3x^2) from the corresponding terms in the dividend (x^4 + 0x^3 - 3x^2). This subtraction is performed term by term:

(x^4 - x^4) = 0
(0x^3 - x^3) = -x^3
(-3x^2 - (-3x^2)) = 0

The result of the subtraction is -x^3 + 0x^2. Next, we bring down the next term from the dividend, which is +4x. This gives us the new polynomial to work with: -x^3 + 0x^2 + 4x. This step is a crucial juncture in the long division process, as it combines the subtraction and the bringing down of the next term, effectively updating the dividend for the next iteration of the algorithm. Precision in the subtraction and careful attention to bringing down the correct term are essential for maintaining the accuracy of the long division.

Step 4: Repeat the Process

We now repeat the process, focusing on the leading term of the new polynomial (-x^3) and the leading term of the divisor (x^2). We ask ourselves, what must we multiply x^2 by to obtain -x^3? The answer is -x. This -x becomes the next term in our quotient, which we add to the quotient obtained in step 1. Now, we multiply -x by the divisor (x^2 + x - 3), which gives us -x^3 - x^2 + 3x. We write this below the corresponding terms of the current polynomial and subtract. The subtraction yields:

(-x^3 - (-x^3)) = 0
(0x^2 - (-x^2)) = x^2
(4x - 3x) = x

The result is x^2 + x. We then bring down the next term from the dividend, which is -3, giving us the new polynomial x^2 + x - 3. This repetition of the division process highlights the iterative nature of the long division algorithm. Each iteration reduces the degree of the polynomial being divided, gradually bringing us closer to the final quotient and remainder.

Step 5: Final Step

We repeat the process one last time. The leading term of the current polynomial (x^2 + x - 3) is x^2, which is the same as the leading term of the divisor. Thus, we multiply the divisor by 1. Adding 1 to the quotient, we multiply 1 by the divisor (x^2 + x - 3), which gives us x^2 + x - 3. Subtracting this from the current polynomial results in:

(x^2 - x^2) = 0
(x - x) = 0
(-3 - (-3)) = 0

Since the remainder is 0, the division is exact. The process concludes when the degree of the remaining polynomial is less than the degree of the divisor or when the remainder is zero. This final step is a culmination of all the previous iterations, resulting in the identification of the last term of the quotient and the determination of the remainder. A zero remainder indicates a perfect division, affirming that the divisor is a factor of the dividend.

The Quotient Polynomial

After performing the polynomial long division, the quotient we obtained is x^2 - x + 1. This signifies that when we divide the polynomial (x^4 - 3x^2 + 4x - 3) by the polynomial (x^2 + x - 3), the result is x^2 - x + 1, with no remainder. The quotient polynomial, x^2 - x + 1, represents the polynomial that, when multiplied by the divisor (x^2 + x - 3), yields the dividend (x^4 - 3x^2 + 4x - 3). This quotient provides valuable information about the relationship between the dividend and the divisor, shedding light on the factors and roots of the polynomials involved. The determination of the quotient is often the primary goal in polynomial division, as it simplifies complex expressions and aids in solving equations and inequalities involving polynomials. The quotient polynomial serves as a key building block in further algebraic manipulations and analyses.

Conclusion

In summary, through the meticulous application of polynomial long division, we have successfully determined that the quotient of (x^4 - 3x^2 + 4x - 3) divided by (x^2 + x - 3) is x^2 - x + 1. The absence of a remainder in this division signifies that (x^2 + x - 3) is a factor of (x^4 - 3x^2 + 4x - 3). This process underscores the importance of polynomial long division as a powerful tool in algebraic manipulation and simplification. The ability to accurately perform polynomial division is crucial for solving a wide range of mathematical problems, including factoring polynomials, solving polynomial equations, and simplifying rational expressions. The step-by-step approach outlined in this article provides a clear and concise method for tackling polynomial division, empowering readers to confidently approach similar problems in the future. Mastering polynomial long division not only enhances one's algebraic skills but also lays a solid foundation for more advanced mathematical concepts.