Dividing Polynomials: Finding the Quotient for ( x^3 - 6x^2 11x - 6 ) รท ( x^2 - 4x 3 )
Abstract: In this article, we will delve into the process of polynomial division, specifically the division of the polynomial ( x^3 - 6x^2 11x - 6 ) by the polynomial ( x^2 - 4x 3 ), to find the quotient. This topic is a fundamental concept in algebra and is essential for many areas of mathematics. We will cover the step-by-step process, including factorization and synthetic division, to solve this problem and similar ones.
Introduction to Polynomial Division
Polynomial division involves finding the quotient when one polynomial is divided by another polynomial. It is a critical skill in algebra and serves as a basis for solving equations and performing various mathematical operations. The division of polynomials can be done using either long division or synthetic division, depending on the nature of the polynomials involved.
Factorizing the Polynomials
To divide ( x^3 - 6x^2 11x - 6 ) by ( x^2 - 4x 3 ), it is helpful to first factorize each polynomial. The factorization process reveals the roots of the polynomials and simplifies the division process.
Factorizing the Numerator:
The numerator ( x^3 - 6x^2 11x - 6 ) can be factored into linear factors.
Given:
Numerator: [ x^3 - 6x^2 11x - 6 (x - 1)(x - 2)(x - 3) ]
To verify, we can expand the product:
[ (x - 1)(x - 2)(x - 3) (x^2 - 3x 2)(x - 3) x^3 - 3x^2 2x - 3x^2 9x - 6 x^3 - 6x^2 11x - 6 ]Factorizing the Denominator:
The denominator ( x^2 - 4x 3 ) can also be factored into linear factors.
Given:
Denominator: [ x^2 - 4x 3 (x - 1)(x - 3) ]
To verify, we can expand the product:
[ (x - 1)(x - 3) x^2 - 3x - x 3 x^2 - 4x 3 ]Polynomial Division
Once the polynomials are factorized, we can proceed to perform the division to find the quotient.
Using Long Division:
The long division method involves dividing the leading terms of the numerator and the denominator, then subtracting the product of the divisor and the result to find the new dividend and repeating the process until the remainder is of lower degree than the divisor.
Divide ( x^3 ) by ( x^2 ) to get ( x ). Multiply ( x ) by ( x^2 - 4x 3 ) to get ( x^3 - 4x^2 3x ). Subtract ( x^3 - 4x^2 3x ) from ( x^3 - 6x^2 11x - 6 ) to get a new dividend: ( -2x^2 8x - 6 ). Repeat the process with the new dividend.The detailed steps are provided in the Python code execution as follows:
[ x^3 - 6x^2 11x - 6 (x - 1)(x - 2)(x - 3) ] [ x^2 - 4x 3 (x - 1)(x - 3) ] [text{Quotient} x - 2 ]Using Synthetic Division:
Alternatively, synthetic division can be used for the polynomial ( x^3 - 6x^2 11x - 6 ) and the divisor ( x^2 - 4x 3 ).
The roots of ( x^2 - 4x 3 ) are ( x 1 ) and ( x 3 ). Perform synthetic division with each root. Compare the results to find the quotient.For ( x 1 ):
Write down the coefficients: 1, -6, 11, -6. Perform the division steps using synthetic division: Result: ( x - 2 ).For ( x 3 ):
Write down the coefficients: 1, -6, 11, -6. Perform the division steps using synthetic division: Result: ( x - 2 ).The detailed synthetic division steps are provided using the following Python code:
```python from sympy import symbols, div, Poly x symbols('x') numerator Poly(x**3 - 6*x**2 11*x - 6, x) denominator Poly(x**2 - 4*x 3, x) quotient, remainder div(numerator, denominator) quotient ``` The output will be:Quotient: ( x - 2 )
Conclusion
In this article, we explored the process of dividing the polynomial ( x^3 - 6x^2 11x - 6 ) by the polynomial ( x^2 - 4x 3 ) by both long division and synthetic division methods. The quotient derived was ( x - 2 ), which we demonstrated through step-by-step factorization and verification. Understanding polynomial division is crucial for solving complex algebraic problems and is a fundamental skill in higher mathematics.