Polynomial Long Division
Polynomial long division is a mathematical technique used to divide one polynomial by another polynomial. It is analogous to long division with integers, but instead of dividing numbers, we divide polynomials. The polynomial long division allows us to simplify and factorize polynomials, find polynomial remainders, and determine polynomial equations.
The polynomial long division allows us to determine if a polynomial is a factor of another polynomial and helps in factorizing polynomials into their irreducible components. Solving polynomial equations, finding rational roots, and performing polynomial interpolation polynomial long division is a fundamental tool.
In this article, we will discuss the definition of polynomial long division, the formula of polynomial long division and we discuss the steps used for finding the polynomial long division. In addition, with the help of examples, we explained the topic.
Polynomial Long Division
Polynomial long division is a method of dividing one polynomial by another polynomial to obtain a quotient and a remainder. It is an algebraic technique used to simplify and factorize polynomials.
A polynomial long division is a method used to divide one polynomial by another polynomial. It is an extended version of the traditional long-division algorithm used for dividing numbers. The polynomial long division allows us to divide polynomials and obtain a quotient and remainder.
Formula
The formula for polynomial long division involves the division of one polynomial (the dividend) by another polynomial (the divisor) to obtain a quotient and a remainder.
The long division formula is:
Dividend = Divisor × Quotient + Remainder
In polynomial long division, we divide the terms of the dividend polynomial by the terms of the divisor polynomial, one term at a time, until the degree of the remaining polynomial is lower than the degree of the divisor. The terms of the quotient polynomial are determined by the quotients obtained during the division process, and the remainder polynomial consists of the terms that cannot be divided further.
Polynomial Long Division: Steps
- The order of the dividend and divisor should be in decreasing order of degree.
- Divide the highest degree term of the dividend by the highest degree term of the divisor to obtain the first term of the quotient.
- Multiply the first term of the quotient with the entire divisor.
- Step 3 result subtracted from the dividend.
- Bring down the next term of the dividend (if any) and append it to the result obtained in step 4.
- Divide the new dividend (the result of step 5) by the divisor.
- Whenever the new dividend is less than the degree of divisor repeat the repeatedly steps 3-6.
- The result is the quotient obtained by adding all the terms of the quotient obtained in each step. The remainder is the last term obtained from the dividend.
Examples of Polynomial Long Division
Below are a few solved examples to understand how to calculate polynomial division problems. You can also try a polynomial long division calculator to get the quotient and remainder of the given problems in a couple of seconds.
Example 1:
Polynomial = 2x3 + x2 + 12x + 10
Divisor= x-1.
Polynomial long division=?
Solution
Given data
P = 2x3 + x2 + 12x + 10
D= x – 1
Step 1: Divide the leading term first, and then multiply the result by the divisor.
2x3/x = 2x2
It implies that the first step is obtained by dividing 2x2.
2x2(x-1) = 2x3-2x2
Remainder= 3x2 + 12x + 10
Step 2: Divide the leading term once more, then multiply the result by the divisor.
3x2/x = 3x
Multiply 3x to the divisor and add the answer to get the reminder.
3x(x-1) = 3x2-3x
Reminder= 15x+10
Step 3: Repeat the previous procedures once again to obtain the third divisional remaining.
15x/x = 15
15 Multiply to the divisor.
15(x-1) = 15x-15
After adding the answer is 25
Q= 2x3 + x2 + 12x + 10
R = 25
Step 5: Mathematically result of the polynomial long division.
2x2+3x + 15
x-1| 2x3 + x2 + 12x + 10
± 2x3 ∓ 2x2
3x2 + 12x+ 10
± 3x2 ∓ 3x
15x +10
± 15x ∓ 15
25
Example 2:
Divide the polynomial f(x) = 5x3 – 3x2 + 2x – 1 and the devisor is g(x) = x +1.
Find the Polynomial long division by using the long division method.
Answer
Given data
f(x) = 5x3 – 3x2 + 2x – 1
g(x) = x +1.
Step 1: Divide the leading term first, then multiply the result by the divisor.
5x3/x = 5x2
It implies that the first step is obtained by dividing 5x2.
5x2(x+1) = 5x3+5x2
Remainder= 8x2 + 2x -1
Step 2: Divide the leading term once more, then multiply the result by the divisor.
8x2/x = 8x
Multiply 8x to the divisor and add the answer to get the reminder.
8x(x+1) = 8x2+8x
Reminder= -6x-1
Step 3: Repeat the previous procedures once again to obtain the third divisional remaining.
-6x/x =-6
Multiply the -6 by the devisor.
-6(x+1) = -6x-6
After adding the answer is 5
Q= 5x3 – 3x2 + 2x – 1
R = 5
Step 5: Mathematically result of the polynomial long division.
5x2+8x -6
X+1| 5x3 – 3x2 + 2x – 1
± 5x3 ± 5x2
-8x2 + 2x- 1
± 8x2 ± 8x
-6x -1
∓6x ∓6
5
Frequently Asked Question
Question 1:
What is the purpose of polynomial long division?
Answer:
Polynomial long division is used to divide one polynomial by another polynomial, allowing us to find the quotient and remainder. It helps simplify and analyze polynomial expressions, factor polynomials, solve equations, and perform other operations involving polynomials.
Question 2:
How do I determine the order of the terms during polynomial long division?
Answer:
When performing polynomial long division, it is essential to arrange the terms of both the dividend and divisor in descending order of their degree (highest power of x to the lowest power). This ensures the correct alignment and simplifies the division process.
Question 3:
Can polynomial long division be used to solve polynomial equations?
Answer:
Yes, polynomial long division can be used as a step in solving polynomial equations. By dividing a polynomial equation by a linear factor, you can reduce the equation to a simpler form and potentially find the roots (solutions) of the equation.
Summary
In this article, we have discussed the definition of polynomial long division, the formula of polynomial long division and we discuss the steps used for finding the polynomial long division. And also, with the help of examples, we explained the topic. After completely understanding this article, anyone can defend this article easily.
Disclaimer: quizsansar.in does not own this book, PDF Materials Images, neither created nor scanned. We just provide Notes already available on the Book and internet. If any way it violates the law or has any issues then kindly mail us: teamshubhamyadav@gmail.com
