Remainder Theorem: Polynomials

Introduction

This tutorial explains the Remainder Theorem and its application in polynomial division. Understanding this theorem is essential for evaluating polynomials and simplifying complex algebraic expressions. We will cover the Division Algorithm, polynomial evaluation, and synthetic division through practical examples.

Step 1: Understanding the Remainder Theorem

• The Remainder Theorem states that when a polynomial ( f(x) ) is divided by ( x - c ), the remainder is ( f(c) ).
• This means that if you want to find the remainder of a polynomial when divided by a linear factor, you can simply evaluate the polynomial at ( c ).
• Example:
  • For ( f(x) = 2x^3 - 3x^2 + 4x - 5 ), to find the remainder when divided by ( x - 2 ):
    • Calculate ( f(2) ):
      • ( f(2) = 2(2)^3 - 3(2)^2 + 4(2) - 5 )
      • ( f(2) = 16 - 12 + 8 - 5 = 7 )
    • The remainder is 7.

Step 2: Applying the Division Algorithm

• The Division Algorithm states that for any polynomial ( f(x) ) and a divisor ( d(x) ), there exist unique polynomials ( q(x) ) (the quotient) and ( r(x) ) (the remainder) such that:
  • ( f(x) = d(x) \cdot q(x) + r(x) )
• Ensure that the degree of ( r(x) ) is less than the degree of ( d(x) ).
• Example:
  • If you divide ( f(x) ) by ( d(x) = x - 2 ), and ( r(x) ) is a constant (degree 0), the remainder can be found using the Remainder Theorem.

Step 3: Performing Synthetic Division

• Synthetic division is a simplified method for dividing polynomials, especially useful when dividing by linear factors.

• Steps to perform synthetic division:

  1. Write down the coefficients of the polynomial.
  2. Write the value of ( c ) (the root of the divisor ( x - c )).
  3. Bring down the leading coefficient.
  4. Multiply ( c ) by the number just brought down and write the result beneath the next coefficient.
  5. Add the column, and repeat the multiplication and addition until all coefficients have been processed.
  6. The last number is the remainder.

• Example:

  • For ( f(x) = 2x^3 - 3x^2 + 4x - 5 ) and dividing by ( x - 2 ):
    • Coefficients are [2, -3, 4, -5].
    • Value of ( c = 2 ).
    • Steps:
      • Bring down 2: [2].
      • Multiply ( 2 ) (c) by ( 2 ): 4, add to -3: [2, 1].
      • Multiply ( 2 ) by 1: 2, add to 4: [2, 1, 6].
      • Multiply ( 2 ) by 6: 12, add to -5: [2, 1, 6, 7].
    • The remainder is 7.

Conclusion

The Remainder Theorem provides a quick way to evaluate polynomials, while the Division Algorithm and synthetic division streamline the process of dividing polynomials. Mastering these concepts is crucial for further studies in algebra and calculus. Practice using these methods with various polynomials to reinforce your understanding and improve your skills in polynomial manipulation.