Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions which includes one or several terms, each of which has a variable raised to a power. Dividing polynomials is a crucial function in algebra which involves finding the remainder and quotient when one polynomial is divided by another. In this blog article, we will investigate the various techniques of dividing polynomials, consisting of long division and synthetic division, and offer examples of how to apply them.
We will also talk about the importance of dividing polynomials and its applications in various fields of mathematics.
Prominence of Dividing Polynomials
Dividing polynomials is an essential operation in algebra which has several utilizations in various fields of mathematics, including calculus, number theory, and abstract algebra. It is used to solve a broad range of problems, including finding 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, that is used to work out the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is used to learn the features of prime numbers and to factorize huge values into their prime factors. It is further utilized to study algebraic structures for instance rings and fields, which are fundamental ideas in abstract algebra.
In abstract algebra, dividing polynomials is used to determine polynomial rings, that are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in many domains of arithmetics, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a method of dividing polynomials that is utilized to divide a polynomial by a linear factor of the form (x - c), where 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) gives 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, using the constant as the divisor, and carrying out a chain of calculations to figure out the quotient and remainder. The outcome is a simplified form of the polynomial which is easier to function with.
Long Division
Long division is an approach of dividing polynomials that is used to divide a polynomial with any other polynomial. The method is based on the reality that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, at which point 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 greatest degree term of the dividend with the highest degree term of the divisor, and subsequently multiplying the result with the entire divisor. The answer is subtracted from the dividend to reach the remainder. The method is recurring until 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 assume 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 streamline 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. 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 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 apply long division to streamline the expression:
First, we divide the largest degree term of the dividend with the highest degree term of the divisor to attain:
6x^2
Next, we multiply the total divisor by the quotient term, 6x^2, to get:
6x^4 - 12x^3 + 6x^2
We subtract this from the dividend to attain 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 recur the process, dividing the largest degree term of the new dividend, 7x^3, by the highest degree term of the divisor, x^2, to achieve:
7x
Then, we multiply the entire divisor with the quotient term, 7x, to achieve:
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)
that streamline to:
10x^2 + 2x + 3
We recur the procedure 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 whole divisor by the quotient term, 10, to obtain:
10x^2 - 20x + 10
We subtract this from the new dividend to obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that simplifies to:
13x - 10
Therefore, the outcome of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We can express f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
In Summary, dividing polynomials is an essential operation in algebra that has multiple utilized in various fields of math. Getting a grasp of the various approaches of dividing polynomials, for instance long division and synthetic division, can guide them in solving complex problems efficiently. Whether you're a learner struggling to get a grasp algebra or a professional working in a field which consists of polynomial arithmetic, mastering the ideas of dividing polynomials is important.
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