Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions which consist of one or several terms, each of which has a variable raised to a power. Dividing polynomials is an important function in algebra that involves finding the remainder and quotient when one polynomial is divided by another. In this blog article, we will examine the various methods of dividing polynomials, including long division and synthetic division, and provide instances of how to utilize them.
We will further talk about the significance of dividing polynomials and its uses in various fields of math.
Significance of Dividing Polynomials
Dividing polynomials is an important function in algebra that has multiple utilizations in various domains of math, including number theory, calculus, and abstract algebra. It is used to work out a extensive range of problems, involving working out the roots of polynomial equations, working out 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 time. The quotient rule of differentiation consists of dividing two polynomials, which is applied 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 huge figures into their prime factors. It is also utilized to study algebraic structures for example fields and rings, that are fundamental concepts in abstract algebra.
In abstract algebra, dividing polynomials is applied to determine polynomial rings, that are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are applied in multiple fields of arithmetics, including algebraic geometry and algebraic number theory.
Synthetic Division
Synthetic division is an approach of dividing polynomials that is applied to divide a polynomial with 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) gives 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 working out a sequence of workings to figure out the quotient and remainder. The outcome is a simplified structure of the polynomial which is straightforward to work with.
Long Division
Long division is an approach of dividing polynomials that is used to divide a polynomial by any other polynomial. The approach is relying 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, next the division of f(x) by g(x) provides us a quotient polynomial of degree n-m and a remainder of degree m-1 or less.
The long division algorithm consists of dividing the greatest degree term of the dividend with the highest degree term of the divisor, and then multiplying the answer by the total divisor. The outcome is subtracted from the dividend to reach the remainder. The method is repeated until the degree of the remainder is lower 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 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. Therefore, we can express f(x) as:
f(x) = (x - 1)(3x^2 + 7x + 2) + 4
Example 2: Long Division
Example 2: Long Division
Let's assume 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 use long division to simplify the expression:
To start with, we divide the largest degree term of the dividend by the highest degree term of the divisor to attain:
6x^2
Next, we multiply the entire divisor with the quotient term, 6x^2, to obtain:
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 simplifies to:
7x^3 - 4x^2 + 9x + 3
We recur the procedure, dividing the largest degree term of the new dividend, 7x^3, with the highest degree term of the divisor, x^2, to obtain:
7x
Next, we multiply the total divisor by the quotient term, 7x, to achieve:
7x^3 - 14x^2 + 7x
We subtract this of the new dividend to obtain the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
that simplifies to:
10x^2 + 2x + 3
We recur the process again, dividing the largest degree term of the new dividend, 10x^2, by the highest degree term of the divisor, x^2, to obtain:
10
Then, we multiply the entire divisor by the quotient term, 10, to obtain:
10x^2 - 20x + 10
We subtract this of the new dividend to obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that simplifies to:
13x - 10
Thus, 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
In conclusion, dividing polynomials is a crucial operation in algebra that has multiple applications in numerous fields of mathematics. Getting a grasp of the different techniques of dividing polynomials, such as long division and synthetic division, could guide them in solving complicated problems efficiently. Whether you're a student struggling to understand algebra or a professional working in a field which consists of polynomial arithmetic, mastering the theories of dividing polynomials is important.
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