Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions that comprises of one or several terms, all of which has a variable raised to a power. Dividing polynomials is an important function in algebra which involves finding the remainder and quotient when one polynomial is divided by another. In this blog, we will explore the different methods of dividing polynomials, consisting of long division and synthetic division, and offer examples of how to apply them.
We will also discuss the significance of dividing polynomials and its applications in various fields of math.
Prominence of Dividing Polynomials
Dividing polynomials is a crucial operation in algebra which has many uses in various domains of arithmetics, involving calculus, number theory, and abstract algebra. It is applied to work out a extensive array of challenges, involving figuring out the roots of polynomial equations, calculating limits of functions, and solving differential equations.
In calculus, dividing polynomials is utilized to work out the derivative of a function, which is the rate of change of the function at any point. The quotient rule of differentiation includes dividing two polynomials, which is utilized to find the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is used to study the features of prime numbers and to factorize large numbers into their prime factors. It is also utilized to study algebraic structures such as rings and fields, that are basic concepts in abstract algebra.
In abstract algebra, dividing polynomials is used to specify polynomial rings, that are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in multiple domains of mathematics, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is an approach of dividing polynomials that is applied to divide a polynomial by a linear factor of the form (x - c), where c is a constant. The approach is founded on the fact that if f(x) is a polynomial of degree n, therefore the division of f(x) by (x - c) offers 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, using the constant as the divisor, and performing a chain of calculations to find the quotient and remainder. The result is a simplified structure of the polynomial that is easier to work with.
Long Division
Long division is an approach of dividing polynomials which is applied to divide a polynomial by any other polynomial. The approach is relying on the fact that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, at which point m ≤ n, next 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 includes dividing the greatest degree term of the dividend by the highest degree term of the divisor, and subsequently multiplying the answer with the total divisor. The result is subtracted of the dividend to get the remainder. The process is repeated as far as 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 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 simplify the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The result of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Therefore, 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 with the polynomial g(x) = x^2 - 2x + 1. We could utilize long division to streamline the expression:
First, we divide the highest degree term of the dividend by the highest degree term of the divisor to attain:
6x^2
Then, we multiply the total divisor with the quotient term, 6x^2, to get:
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 simplifies to:
7x^3 - 4x^2 + 9x + 3
We recur the procedure, dividing the largest degree term of the new dividend, 7x^3, by the largest degree term of the divisor, x^2, to get:
7x
Next, we multiply the whole divisor with 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 streamline to:
10x^2 + 2x + 3
We recur the method again, dividing the largest degree term of the new dividend, 10x^2, by the largest degree term of the divisor, x^2, to get:
10
Subsequently, we multiply the total divisor by the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this of the new dividend to get the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
which simplifies to:
13x - 10
Therefore, the result 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 multiple applications in multiple domains of mathematics. Comprehending the various methods of dividing polynomials, such as long division and synthetic division, can help in figuring out intricate challenges efficiently. Whether you're a student struggling to get a grasp algebra or a professional operating in a domain that includes polynomial arithmetic, mastering the ideas of dividing polynomials is essential.
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