Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions which comprises of one or several terms, all of which has a variable raised to a power. Dividing polynomials is an essential function in algebra that involves figuring out the quotient and remainder as soon as one polynomial is divided by another. In this blog article, we will investigate the various techniques of dividing polynomials, consisting of synthetic division and long division, and provide instances of how to apply them.
We will further talk about the importance of dividing polynomials and its uses in multiple domains of math.
Importance of Dividing Polynomials
Dividing polynomials is an important function in algebra which has many uses in various domains of math, including calculus, number theory, and abstract algebra. It is applied to figure out a wide range of problems, including working out the roots of polynomial equations, figuring out limits of functions, and working out differential equations.
In calculus, dividing polynomials is utilized to find the derivative of a function, which is the rate of change of the function at any time. The quotient rule of differentiation includes dividing two polynomials, that is used to find the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is used to study the characteristics of prime numbers and to factorize huge numbers into their prime factors. It is further utilized to learn algebraic structures such as rings and fields, which are fundamental theories in abstract algebra.
In abstract algebra, dividing polynomials is used to determine polynomial rings, which are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are applied in various fields of mathematics, including algebraic geometry and algebraic number theory.
Synthetic Division
Synthetic division is a technique of dividing polynomials which is applied to divide a polynomial by a linear factor of the form (x - c), at point which c is a constant. The technique is based on the fact that if f(x) is a polynomial of degree n, then 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 involves writing the coefficients of the polynomial in a row, utilizing the constant as the divisor, and working out a chain of workings to find the quotient and remainder. The result is a streamlined form of the polynomial which is easier to work with.
Long Division
Long division is a technique of dividing polynomials that is used to divide a polynomial by another polynomial. The method is on the basis the reality that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, where m ≤ n, then 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 by the whole divisor. The answer is subtracted from the dividend to get the remainder. The procedure is repeated as far as the degree of the remainder is lower compared to the degree of the divisor.
Examples of Dividing Polynomials
Here are few examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's say we have to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 with the linear factor (x - 1). We could use synthetic division to streamline the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The answer of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Thus, 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 need to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 by the polynomial g(x) = x^2 - 2x + 1. We can utilize long division to streamline the expression:
First, we divide the largest degree term of the dividend with the largest degree term of the divisor to attain:
6x^2
Subsequently, we multiply the total divisor by 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)
that streamlines to:
7x^3 - 4x^2 + 9x + 3
We repeat the method, dividing the highest degree term of the new dividend, 7x^3, by the highest degree term of the divisor, x^2, to achieve:
7x
Next, we multiply the whole divisor by the quotient term, 7x, to achieve:
7x^3 - 14x^2 + 7x
We subtract this from the new dividend to get the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
which simplifies to:
10x^2 + 2x + 3
We recur the method again, dividing the highest degree term of the new dividend, 10x^2, with the largest degree term of the divisor, x^2, to get:
10
Then, we multiply the whole divisor with the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this from the new dividend to achieve the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that streamlines to:
13x - 10
Thus, the outcome 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 an important operation in algebra which has several applications in numerous fields of mathematics. Getting a grasp of the different techniques of dividing polynomials, for example long division and synthetic division, could help in figuring out complicated problems efficiently. Whether you're a learner struggling to get a grasp algebra or a professional operating in a field which includes polynomial arithmetic, mastering the concept of dividing polynomials is important.
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