Division Algorithm for Linear Divisors
Division algorithm for polynomials provides us with a step-by-step procedure for evaluating the division of a polynomial by another polynomial. Here in specific let us consider to divide a polynomial with a linear divisor. A linear divisor is yet another polynomial whose degree is one. We call it the division algorithm, where the word algorithm means a step-by-step procedure for calculating or evaluating the results and finding the solutions. In this lesson we will learn about division algorithm for linear divisors and in the process learn to determine the quotient polynomial and the (constant) remainder. Stay tuned to learn more!!!
Division Algorithm
The division algorithm states that dividend can be expressed as sum of remainder to the product of quotient and divisor. It is given as Dividend = (Divisor × Quotient) + Remainder.
Similarly in the case of polynomials, the division algorithm for polynomials is used to determine the quotient polynomial and the (constant) remainder. Let the p(x), q(x), g(x) and r(x) represent, dividend polynomial, quotient polynomial, divisor polynomial and remainder respectively. Hence, the division algorithm for them is given as, p(x) = g(x) × q(x) + r(x).
Polynomial Long Division
The steps to divide polynomials with linear divisors are:
- First we arrange the terms of the dividend and divisor in descending order of their powers.
- The first term of the quotient polynomial is determined by dividing the highest degree term of the dividend and largest degree term of the divisor.
- The next subsequent term of the quotient by dividing the highest term of the new dividend obtained from above steps with the largest term of the divisor.
- These steps are repeated until we obtain the degree of remainder less than degree of divisor.
Let us take a concrete example. Suppose that the dividend a(x) and the divisor b(x) are given by
a(x)= 2x3 - x2 + x - 1
b(x) = x + 7
- Now, we think of that term with which we multiply the first term of b(x) (which is x) to generate the first term of a(x). It is written as, x × ? = 2x3
- Obviously, the multiplier is 2x2. Now, we multiply the divisor by this multiplier which we have figured out, and write the result below the dividend, so that terms of the same degree align with each other.
- Next, we subtract this new polynomial from the original dividend, and obtain our dividend for the next step of the algorithm.
- Once we have this, we figure out the next multiplier: that term with which we multiply the first term of b(x) (which is x) to generate the highest degree term of the new dividend.
- That multiplier is - 15x.
- Finally, the last multiplier will be 106, and in the last step of the algorithm.
Thus, the quotient polynomial and the remainder are,
q(x) = 2x2 - 15x + 106
r = - 743
Topics Related to Division Algorithm for Linear Divisors
Solved Examples on Division Algorithm for Linear Divisors
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Example 1: Consider the following two polynomials:
a(x)= 6x4 - x3 + 2x2 - 7x + 2
b(x)=2x + 3Find the quotient polynomial and the remainder when a(x) is divided by b(x).
Solution: We first work out this problem in the following way. In this solution, M1, M2, etc, are the multipliers with which we multiply the divisor at each successive step of the algorithm. M1 is the first multiplier, M2 is the second multiplier, and so on:
Thus, the quotient polynomial and the remainder are:
q(x) = 3x3 + 4x2 - 5x + 11
r = -31 -
Example 2: Consider the following two polynomials:
a(x) = x3 - x2 + x - 1
b(x) = 2x + 1Find the quotient polynomial and the remainder when the polynomials are divided.
Solution: The polynomials can be divided as below. M1 is the first multiplier, M2 is the second multiplier, and so on:
Thus, we have,
q(x) = (1/2)x2 - (3/4)x + (7/8)
r = - (15/8)
FAQs on Division Algorithm for Linear Divisors
What is Division Algorithm Formula?
The division algorithm formula is, Dividend = (Divisor × Quotient) + Remainder.
How Do You Divide Polynomials with Linear Divisors?
The steps to divide polynomials with linear divisors are:
- First we arrange the terms of the dividend and divisor in descending order of their powers.
- The first term of the quotient polynomial is determined by dividing the highest degree term of the dividend and the largest degree term of the divisor.
- The next subsequent term of the quotient by dividing the highest term of the new dividend obtained from above steps with the largest term of the divisor.
What are Linear Divisors?
The linear divisors are linear polynomials which are written in a general form ax+b having their degree as 1.
Are Polynomials Closed Under Division?
No, polynomials are not closed under division.
What is Division Algorithm for Polynomials?
If p(x), q(x), g(x) and r(x) represent, dividend polynomial, quotient polynomial, divisor polynomial and remainder respectively, the division algorithm for them is given as, p(x) = g(x) × q(x) + r(x).