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Polynomials

Value of Polynomial and Division Algorithm

Arithmetic operations like addition, subtraction, multiplication and division play a huge and most basic rule in Mathematics. Maths is made by these operations. All other operations go easy with the polynomials except the division operation, which gets complex when dealt with polynomials. But this section will explain to you the division of polynomials and the division algorithm related to it, from basics.

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So, what’s the basic formula we are learning from the day we solved our first division problem? This is:

Dividend = Quotient × Divisor + Remainder

Example: Divide the polynomial 2x2+3x+1 by polynomial x+2.
Solution: Divisor= x+2
Dividend=2x2 + 3x + 1
Note: Put the dividend under the division sign and divisor outside the sign.

Division Algorithm

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Steps for Division of Polynomials

  • Step 1: Firstly, Arrange the divisor as well as dividend individually in decreasing order of their degree of terms.
  • Step 2: In case of division we seek to find the quotient. To find the very first term of the quotient, divide the first term of the dividend by the highest degree term in the divisor. In the current case,

2x2/x = 2x.

  • Step 3: Write 2x in place of the quotient.
  • Step 4: Multiply the divisor by the quotient obtained. Put the product underneath the dividend.
  • Step 5: Subtract the product obtained as happens in case of a division operation.
  • Step 6: Write the result obtained after drawing another bar to separate it from prior operations performed.
  • Step 7: Bring down the remaining terms of the dividend.
  • Step 8: Again divide the dividend by the highest degree term of the remaining divisor. Follow the same prior procedure until either the remainder becomes zero or its degree is less than the degree of the divisor.
  • Step 9: At this stage, the quotient obtained is our answer.

Quotient Obtained = 2x + 1

Note: Division Algorithms for Polynomials is same as the Long Division Algorithm In Polynomials

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Division Algorithm For Polynomials

Division algorithm for polynomials states that, suppose f(x) and g(x) are the two polynomials, where g(x)≠0, we can write:

f(x) = q(x) g(x) + r(x)

which is same as the Dividend = Divisor * Quotient + Remainder and where r(x) is the remainder polynomial and is equal to 0 and degree r(x) < degree g(x).

Verification of Division Algorithm

Take the above example and verify it.

Divisor = x+2
Dividend = 2x2 + 3x + 1
Quotient = 2x – 1
Remainder = 0

Applying the Algorithm:

2x2 + 3x + 1 = (x + 2) (2x + 1) + 0
2x2 + 3x + 1 = 2x2 + 3x + 1 

Hence verified.

Finding Factors of Polynomials with Division Algorithm

Long division algorithm is used to find out factors of polynomials of degree greater than equal to two. We’ll be describing the steps to find out the factors along with an example.

Example: Find roots of cubic polynomial P(x)=3x3 – 5x2 – 11x – 3

Solution

  • Step 1: Use the factor theorem to find a factor of the polynomial.
  • Step 2: First divide the whole equation by the coefficient of the highest degree term of the dividend.

P(x)=3x3 – 5x2 – 11x – 3

On dividing the whole equation by 3,

P(x) =x3 – (5/3)x2 – (11/3)x – 1

  • Step 3: Find out factors of the constant term so obtained. In the present case, factors of the constant term are 1 and -1.
  • Step 4: Put the value of x in P(x) = 3x3 – 5x2 – 11x – 3   equal to 1 and find the remainder. Again put the value of remainder equal to -1 in and find the remainder using remainder theorem. Find the value of x for which remainder is zero for the cubic polynomial.

P (1) = 3(1)3 – 5(1)2 – 11(1) – 3  = -16

P(-1) = 3( -1 )3 – 5( -1 )2 – 11( -1 ) – 3 =0

  • Step 5: Remainder is zero for x = -1. So, (x + 1) is a root of the polynomial.
  • Step 6: By Division Algorithm, find out the quotient. It comes out:  3x– 8x – 3
  • Step 7: Now, Quotient = 3x– 8x – 3

Dividend = (Divisor) * (Quotient) + Remainder

In present case,

3x3 – 5x2 – 11x – 3 = (x + 1) (3x– 8x – 3) + 0

By factorizing the quadratic polynomial we shall be able to find out remaining factors of the cubic polynomial.

  • Step 8: Break middle term in terms of a pair of numbers such that its product is equal to -9 and summation equal to -3.
  • Step 9: On factorizing,  possible pair of number satisfying both conditions is (-9, 1). Breaking the middle term,

f(x) = 3x– 8x – 3

= 3x– 9x + x – 3

  • Step 10: Form pairs of terms and factor out GCD of the two pairs separately. Then again factor out GCD of the remaining two products.
  • Step 11:

f(x) = 3x– 8x – 3 = 3x– 9x + x – 3

= 3x(x – 3) + 1(x – 3) = (x – 3)(3x + 1)

Now,

3x3 – 5x2 – 11x – 3 = (x + 1) (3x– 8x – 3) + 0

= (x + 1) ( x – 3)(3x + 1)

Factors of cubic polynomial are -1, 3 and -1/3.

Solved Example for You

Question 1: What is the division algorithm formula?

Answer: It states that for any integer, a and any positive integer b, there exists a unique integer q and r such that a = bq + r. Here r is greater than or equal to 0 and less than b. Moreover, a is the dividend, b is the divisor, q is the quotient and r is the remainder.

Question 2: Explain Euclid’s division algorithm?

Answer: It refers to a technique to compute the Highest Common Factor (HCF) of two given positive integers. In addition, let us remind you that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.

Question 3: How does the division algorithm work?

Answer: It refers to an algorithm that gives two integers a and b, and when we compute their quotient and/ or remainder the result of Euclidean division. In addition, we apply some of them by hand, whereas digital circuit designs and software employ others.

Question 4: State the main difference between Lemma and algorithm?

Answer: A Lemma refers to a proven statement that we use to prove another statement. On the other hand, an algorithm refers to a series of well-defined steps that gives a procedure for solving a type of problem.

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