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 a crucial working in algebra that includes finding the quotient and remainder as soon as one polynomial is divided by another. In this blog, we will explore the different approaches of dividing polynomials, involving synthetic division and long division, and give examples of how to apply them.
We will also discuss the importance of dividing polynomials and its utilizations in different fields of math.
Significance of Dividing Polynomials
Dividing polynomials is a crucial operation in algebra which has several applications in many domains of mathematics, including calculus, number theory, and abstract algebra. It is applied to figure out a wide range of challenges, involving finding the roots of polynomial equations, working out limits of functions, and solving differential equations.
In calculus, dividing polynomials is utilized 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 consists of dividing two polynomials, that is utilized to work out the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is utilized to learn the properties of prime numbers and to factorize huge figures into their prime factors. It is also applied to learn algebraic structures for instance rings and fields, that are fundamental concepts in abstract algebra.
In abstract algebra, dividing polynomials is utilized to determine polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are used in various fields of arithmetics, including algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a method of dividing polynomials which is applied to divide a polynomial by a linear factor of the form (x - c), where c is a constant. The approach is on the basis of the fact that if f(x) is a polynomial of degree n, then 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, applying the constant as the divisor, and carrying out a chain of calculations to figure out the quotient and remainder. The answer is a streamlined structure of the polynomial which is simpler to work with.
Long Division
Long division is a method of dividing polynomials which is applied to divide a polynomial by any other polynomial. The technique 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, 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 includes dividing the highest degree term of the dividend with the highest degree term of the divisor, and then multiplying the answer with the whole divisor. The result is subtracted of the dividend to obtain the remainder. The procedure is repeated as far as the degree of the remainder is less compared to the degree of the divisor.
Examples of Dividing Polynomials
Here are some examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's say we need to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We can apply synthetic division to simplify 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. Thus, 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 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 can apply long division to simplify the expression:
To start with, we divide the largest degree term of the dividend with the highest degree term of the divisor to obtain:
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 streamlines to:
7x^3 - 4x^2 + 9x + 3
We recur the procedure, dividing the largest degree term of the new dividend, 7x^3, by the highest degree term of the divisor, x^2, to achieve:
7x
Subsequently, we multiply the entire divisor by the quotient term, 7x, to get:
7x^3 - 14x^2 + 7x
We subtract this from the new dividend to achieve the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
that simplifies to:
10x^2 + 2x + 3
We repeat the process 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
Then, we multiply the entire divisor with 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 streamlines to:
13x - 10
Hence, 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
In Summary, dividing polynomials is an important operation in algebra which has multiple utilized in various fields of math. Understanding the different approaches of dividing polynomials, for example synthetic division and long division, could guide them in working out intricate problems efficiently. Whether you're a learner struggling to comprehend algebra or a professional working in a domain which includes polynomial arithmetic, mastering the theories of dividing polynomials is crucial.
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