Division Algorithm Problems and Solutions

Polynomials are made up of algebraic expressions with different degrees. Degree-one polynomials are called linear polynomials, degree-two are called quadratic and degree-three are called cubic polynomials. Zeros of these polynomials are the points where these polynomials become zero. Sometimes it happens that we have some zeros of the polynomials, we need to find the other zeros. For example, let’s assume a polynomial p(x) = x³ – 3x² – x + 3, and we know that one of the zeros is 1. Then x – 1 must be a factor of this polynomial. The goal is to find the other two zeros. In such cases, the division algorithm helps us. 

Division Algorithm for Polynomials

Let’s have two polynomials p(x) and g(x), and g(x) ≠ 0. Now we can find two polynomials q(x) and r(x) such that, 

p(x) = q(x) x g(x) + r(x), 

Here, either r(x) = 0 or degree of r(x) < degree of g(x). This is called a polynomial division algorithm for polynomials. 

Dividend = Quotient x Divisor + Remainder

Steps for long division: 

Now let’s see with an example, how to divide two polynomials, Let’s say we have p(x) = 2x² + 4x + 1 and g(x) = x + 1. 

• We will stop this process when the remainder becomes zero, or its degree becomes less than the divisor.
• Quotients’ first term is obtained by dividing the highest order term of the dividend with the highest degree term of the divisor.
• For the second term, divide the highest degree term of the new dividend obtained as the remainder by the highest degree term of the divisor.
• Continue the steps until the condition mentioned in Step 1 is met.

Notice that in this example, q(x) = 2x + 2 and r(x) = -1. 

Sample Problems

Question 1: Given the polynomial p(x) = x² + x +5 and g(x) = x +2. Find the value of q(x) and r(x). 

Solution: 

Using the steps mentioned above. On dividing p(x) with g(x) we get, 

q(x) = x – 1 and r(x) = 7. 

Question 2: Given the polynomial p(x) = x³ + x + 6x² + 4 and g(x) = x² + 1. Find the value of q(x) and r(x). 

Solution: 

Using the steps mentioned above. On dividing p(x) with g(x) we get, 

So, here q(x) = x + 6, and r(x) = -2. 

Question 3: Given the polynomial p(x) = x⁴ + x + 6x² + 4 and g(x) = x² + 1. Find the value of q(x) and r(x). 

Solution: 

Using the steps mentioned above. On dividing p(x) with g(x) we get, 

So, here q(x) = x² + 5, and r(x) = x -1. 

Question 4: Given the polynomial x⁴ – 1. We know two of the roots are -1, 1. Find the other two roots if they exist. 

Solution: 

We know that the two roots are -1 and 1. 

So, x -1 and x + 1 are the factors of the given polynomial. Then, (x – 1)(x +1) is also a factor of the polynomial. 

(x – 1) (x + 1) = x² – 1

We see that quotient is x² + 1. The roots of this polynomial will be the roots of the equation. 

x² + 1 = 0

⇒ x² = -1 

This equation cannot have real roots, thus roots don’t exist for this polynomial. 

Question 5: Given the polynomial p(x) = x⁵ + 8x³ – 6x⁴ + 5x² + 10x + 8 and g(x) = x² + 10x -5. Find q(x) and r(x).

Solution: 

Using the same steps as done in previous questions, 

q(x) = x³ – 16x² + 173x – 1805 

r(x) = 18295x – 9017

Question 6: For the same given the polynomial p(x) = x⁵ + 8x³ – 6x⁴ + 5x² + 10x + 8 and g(x) = x + 5. Find q(x) and r(x).

Solution: 

Here, q(x) = x⁴ – 11x³ + 63x² – 310x + 1560 

r(x) = -7792 

Question 7: For the same given the polynomial p(x) = x⁵ – 6x⁴ + 5x² + 8 and g(x) = x + 2. Find q(x) and r(x).

Solution: 

q(x) = x⁴ – 8x³ + 16x² -27x + 54 

g(x) = -100

Division Algorithm Practice Problems

Problem 1: Find the quotient and remainder when 12345 is divided by 97.

Problem 2: Given that when 3456 is divided by 89 the remainder is 56. Find the quotient.

Problem 3: Divide 6789 by 123 and express the result in terms of quotient and remainder.

Problem 4: If a divided by 25 gives a quotient of 37 and a remainder of 8 find the value of a.

Problem 5: Determine the quotient and remainder when 98765 is divided by 543.

Problem 6: Find the remainder when 1001 is divided by 7 and 11. Use the results to find the quotient for each case.

Problem 7: When 2024 is divided by 40 the remainder is r. Find r and express 2024 in terms of the quotient and remainder.

Problem 8: If 5678 is divided by 54 what is the remainder? Also, find the quotient.

Problem 9: Find the quotient and remainder when 4321 is divided by 89.

Problem 10: Given that the quotient of 8765 divided by 89 is 98 find the remainder.

Conclusion

The Division Algorithm is a key concept in number theory that provides the systematic way to the divide integers and find the quotient and remainder. Understanding and applying this algorithm is crucial for the solving problems involving the division and modular arithmetic. The ability to the find the quotient and remainder efficiently supports the various applications in the mathematics and computer science including the algorithm design and data processing.