Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are mathematical expressions that includes one or more terms, each of which has a variable raised to a power. Dividing polynomials is an essential working in algebra which includes working out the quotient and remainder once one polynomial is divided by another. In this blog, we will examine the different methods of dividing polynomials, involving synthetic division and long division, and provide examples of how to use them.
We will further talk about the importance of dividing polynomials and its uses in multiple fields of mathematics.
Importance of Dividing Polynomials
Dividing polynomials is an important function in algebra that has several uses in diverse fields of arithmetics, including calculus, number theory, and abstract algebra. It is utilized to figure out a wide range of problems, consisting of working out the roots of polynomial equations, working out limits of functions, and calculating 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 utilized to work out the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is applied to learn the properties of prime numbers and to factorize huge figures into their prime factors. It is further used to study algebraic structures for instance rings and fields, that are basic concepts in abstract algebra.
In abstract algebra, dividing polynomials is used to specify polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are applied in various domains of math, including algebraic number theory and algebraic geometry.
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), at point which c is a constant. The method is on the basis of the fact that if f(x) is a polynomial of degree n, subsequently 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 involves writing the coefficients of the polynomial in a row, using the constant as the divisor, and carrying out a sequence of calculations to figure out the quotient and remainder. The answer is a simplified form of the polynomial that is easier to function with.
Long Division
Long division is an approach of dividing polynomials that is utilized to divide a polynomial by any other polynomial. The technique 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, 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 includes dividing the highest degree term of the dividend with the highest degree term of the divisor, and then multiplying the outcome by the whole divisor. The answer is subtracted from the dividend to obtain the remainder. The method is repeated as far as the degree of the remainder is lower 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 say we have to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by 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 answer 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 have 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 utilize long division to simplify the expression:
First, we divide the largest degree term of the dividend by the highest degree term of the divisor to obtain:
6x^2
Then, we multiply the entire divisor by the quotient term, 6x^2, to get:
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 recur the procedure, dividing the largest degree term of the new dividend, 7x^3, with the largest degree term of the divisor, x^2, to achieve:
7x
Then, we multiply the entire divisor with 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 streamline to:
10x^2 + 2x + 3
We repeat the procedure again, dividing the largest degree term of the new dividend, 10x^2, with the highest degree term of the divisor, x^2, to obtain:
10
Then, we multiply the entire 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 streamlines 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 state f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
In Summary, dividing polynomials is a crucial operation in algebra which has many applications in numerous fields of math. Getting a grasp of the various methods of dividing polynomials, such as long division and synthetic division, can help in figuring out complex challenges efficiently. Whether you're a student struggling to get a grasp algebra or a professional working in a domain that involves polynomial arithmetic, mastering the ideas of dividing polynomials is crucial.
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