Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are mathematical expressions which includes one or more terms, each of which has a variable raised to a power. Dividing polynomials is an important operation in algebra that includes figuring out the quotient and remainder when one polynomial is divided by another. In this article, we will examine the different methods of dividing polynomials, involving long division and synthetic division, and offer instances of how to use them.
We will also discuss the importance of dividing polynomials and its applications in multiple domains of math.
Prominence of Dividing Polynomials
Dividing polynomials is a crucial operation in algebra which has multiple uses in many fields of arithmetics, including calculus, number theory, and abstract algebra. It is used to work out a wide spectrum of problems, including figuring out the roots of polynomial equations, calculating limits of functions, and working out differential equations.
In calculus, dividing polynomials is applied to work out the derivative of a function, that is the rate of change of the function at any moment. The quotient rule of differentiation includes dividing two polynomials, which is used to work out the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is applied to study the features of prime numbers and to factorize large values into their prime factors. It is further used to learn algebraic structures such as rings and fields, that are rudimental theories in abstract algebra.
In abstract algebra, dividing polynomials is applied to define polynomial rings, that are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in many fields of math, including algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a technique of dividing polynomials which is utilized to divide a polynomial by a linear factor of the form (x - c), at point which c is a constant. The method is based on the fact that if f(x) is a polynomial of degree n, subsequently the division of f(x) by (x - c) provides a quotient polynomial of degree n-1 and a remainder of f(c).
The synthetic division algorithm consists of writing the coefficients of the polynomial in a row, applying the constant as the divisor, and performing a series of calculations to work out the quotient and remainder. The result is a streamlined structure of the polynomial that is simpler to work with.
Long Division
Long division is a technique of dividing polynomials that is utilized to divide a polynomial by another polynomial. The technique is based on the reality that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, where m ≤ n, subsequently the division of f(x) by g(x) gives a quotient polynomial of degree n-m and a remainder of degree m-1 or less.
The long division algorithm involves dividing the highest degree term of the dividend with the highest degree term of the divisor, and further multiplying the outcome with the entire divisor. The answer is subtracted of the dividend to get the remainder. The method is recurring until the degree of the remainder is less than the degree of the divisor.
Examples of Dividing Polynomials
Here are few examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's assume we want to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 with the linear factor (x - 1). We could apply synthetic division to streamline the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The outcome of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Hence, we can state f(x) as:
f(x) = (x - 1)(3x^2 + 7x + 2) + 4
Example 2: Long Division
Example 2: Long Division
Let's say we have to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 with the polynomial g(x) = x^2 - 2x + 1. We could use long division to streamline the expression:
To start with, we divide the highest degree term of the dividend by the highest degree term of the divisor to get:
6x^2
Subsequently, we multiply the total divisor with the quotient term, 6x^2, to attain:
6x^4 - 12x^3 + 6x^2
We subtract this from the dividend to obtain the new dividend:
6x^4 - 5x^3 + 2x^2 + 9x + 3 - (6x^4 - 12x^3 + 6x^2)
which streamlines to:
7x^3 - 4x^2 + 9x + 3
We recur the procedure, dividing the largest degree term of the new dividend, 7x^3, with the largest degree term of the divisor, x^2, to get:
7x
Subsequently, we multiply the entire divisor with the quotient term, 7x, to obtain:
7x^3 - 14x^2 + 7x
We subtract this of the new dividend to achieve the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
which streamline to:
10x^2 + 2x + 3
We repeat the process again, dividing the largest degree term of the new dividend, 10x^2, with the largest degree term of the divisor, x^2, to get:
10
Next, we multiply the total divisor by the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this from the new dividend to obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
which simplifies to:
13x - 10
Thus, the answer of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We could state f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
Ultimately, dividing polynomials is a crucial operation in algebra that has multiple utilized in multiple domains of mathematics. Understanding the different methods of dividing polynomials, for example synthetic division and long division, could help in working out intricate problems efficiently. Whether you're a student struggling to understand algebra or a professional working in a field which involves polynomial arithmetic, mastering the theories of dividing polynomials is essential.
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