Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions that comprises of one or several terms, each of which has a variable raised to a power. Dividing polynomials is a crucial function in algebra that includes figuring out the remainder and quotient once one polynomial is divided by another. In this blog, we will explore the different methods of dividing polynomials, including long division and synthetic division, and give examples of how to use them.
We will also discuss the significance of dividing polynomials and its applications in multiple domains of mathematics.
Importance of Dividing Polynomials
Dividing polynomials is a crucial operation in algebra that has multiple uses in many fields of arithmetics, consisting of calculus, number theory, and abstract algebra. It is applied to solve a wide range of challenges, consisting of finding the roots of polynomial equations, figuring out limits of functions, and calculating differential equations.
In calculus, dividing polynomials is utilized to find the derivative of a function, that is the rate of change of the function at any point. The quotient rule of differentiation includes dividing two polynomials, which is used to work out the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is applied to study the features of prime numbers and to factorize huge numbers into their prime factors. It is further used to study algebraic structures for example rings and fields, that are basic 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 math, including algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is an approach of dividing polynomials that is used 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, subsequently 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 sequence of calculations to find the remainder and quotient. The answer is a simplified structure of the polynomial which is easier to function with.
Long Division
Long division is a method of dividing polynomials which is used to divide a polynomial by another polynomial. The technique is founded 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, 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 greatest degree term of the dividend with the highest degree term of the divisor, and subsequently multiplying the outcome by the total divisor. The answer is subtracted from the dividend to get the remainder. The procedure is repeated until the degree of the remainder is less in comparison to the degree of the divisor.
Examples of Dividing Polynomials
Here are some examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's assume we want 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 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 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 get:
6x^2
Then, 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 get 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 largest degree term of the new dividend, 7x^3, with the largest degree term of the divisor, x^2, to obtain:
7x
Then, we multiply the total divisor by the quotient term, 7x, to get:
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 repeat the procedure again, dividing the highest degree term of the new dividend, 10x^2, by the highest degree term of the divisor, x^2, to obtain:
10
Next, we multiply the whole divisor by the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this from the new dividend to get the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
which simplifies 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 express 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 that has many applications in numerous fields of math. Understanding the different techniques of dividing polynomials, for instance synthetic division and long division, could support in solving complicated challenges efficiently. Whether you're a learner struggling to comprehend algebra or a professional working in a field that involves polynomial arithmetic, mastering the ideas of dividing polynomials is crucial.
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