Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are math expressions which comprises of one or more terms, all of which has a variable raised to a power. Dividing polynomials is an important working in algebra which involves figuring out the quotient and remainder when one polynomial is divided by another. In this blog, we will examine the different methods of dividing polynomials, including synthetic division and long division, and give examples of how to apply them.
We will further discuss the importance of dividing polynomials and its utilizations in various domains of mathematics.
Significance of Dividing Polynomials
Dividing polynomials is an essential operation in algebra that has many utilizations in various fields of arithmetics, involving calculus, number theory, and abstract algebra. It is used to figure out a extensive array of problems, involving finding the roots of polynomial equations, working out limits of functions, and working out differential equations.
In calculus, dividing polynomials is utilized to figure out the derivative of a function, which 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 figure out the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is applied to study the properties of prime numbers and to factorize large numbers into their prime factors. It is also used to learn algebraic structures such as rings and fields, which are fundamental ideas in abstract algebra.
In abstract algebra, dividing polynomials is used to determine polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are applied in various fields of math, comprising of algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a method of dividing polynomials which is utilized to divide a polynomial by a linear factor of the form (x - c), at point which 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, using the constant as the divisor, and carrying out a sequence of workings to figure out the quotient and remainder. The result is a streamlined structure of the polynomial that is straightforward to function with.
Long Division
Long division is an approach of dividing polynomials which is used to divide a polynomial with any other polynomial. The technique is founded 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, subsequently the division of f(x) by g(x) offers uf 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 further multiplying the result by the total divisor. The outcome is subtracted of the dividend to obtain the remainder. The method is repeated until the degree of the remainder is lower in comparison to 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 need to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We could utilize 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 say we want 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 streamline the expression:
To start with, we divide the highest degree term of the dividend with 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)
that streamlines to:
7x^3 - 4x^2 + 9x + 3
We recur the process, dividing the highest degree term of the new dividend, 7x^3, by the largest degree term of the divisor, x^2, to achieve:
7x
Subsequently, we multiply the total divisor by the quotient term, 7x, to obtain:
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)
which streamline to:
10x^2 + 2x + 3
We recur the process again, dividing the highest degree term of the new dividend, 10x^2, with the highest degree term of the divisor, x^2, to get:
10
Then, we multiply the whole divisor by the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this of the new dividend to obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that 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 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 several uses in numerous domains of mathematics. Comprehending the different methods of dividing polynomials, such as long division and synthetic division, can 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 ideas of dividing polynomials is important.
If you desire help understanding dividing polynomials or any other algebraic concept, think about reaching out to Grade Potential Tutoring. Our expert teachers are accessible remotely or in-person to provide personalized and effective tutoring services to support you be successful. Call us today to plan a tutoring session and take your mathematics skills to the next level.
