Mathematical Operations On Polynomials

Unit: Polynomials

Mathematical Operations on Polynomials

Polynomials can be manipulated using various mathematical operations. Understanding these operations is crucial for simplifying expressions, solving equations, and performing more advanced algebraic manipulations.

Addition of Polynomials

• Definition: The sum of two polynomials is found by adding the corresponding coefficients of like terms.
• Method:
  1. Align the polynomials by their degree.
  2. Add the coefficients of like terms.
• Example:

(3𝑥²+2𝑥+1) +(5𝑥²−3𝑥+4) =(3𝑥²+5𝑥²) +(2𝑥−3𝑥) +(1+4) =8𝑥²−𝑥+5.

Subtraction of Polynomials

• Definition: The difference of two polynomials is found by subtracting the corresponding coefficients of like terms.
• Method:
  1. Align the polynomials by their degree.
  2. Subtract the coefficients of like terms.
• Example:

(3𝑥²+2𝑥+1) −(5𝑥²−3𝑥+4) =(3𝑥²−5𝑥²) +(2𝑥−(−3𝑥)) +(1−4) =−2𝑥²+5𝑥−3.​

Multiplication of Polynomials

• Definition: The product of two polynomials is found by distributing each term of the first polynomial to each term of the second polynomial.
• Method:
  1. Multiply each term in the first polynomial by each term in the second polynomial.
  2. Combine like terms.
• Example:

(2𝑥+3) (𝑥²−𝑥+4) =2𝑥(𝑥²−𝑥+4) +3(𝑥²−𝑥+4)

=2𝑥³−2𝑥²+8𝑥+3𝑥²−3𝑥+12

=2𝑥³+𝑥²+5𝑥+12.

Division of Polynomials

• Definition: The quotient of two polynomials is found by dividing the terms of the dividend polynomial by the divisor polynomial.
• Method (Long Division):
  1. Divide the leading term of the dividend by the leading term of the divisor.
  2. Multiply the entire divisor by this quotient term and subtract from the dividend.
  3. Repeat the process with the resulting polynomial until the degree of the remainder is less than the degree of the divisor.
• Example: Divide 2𝑥³−3𝑥²+4𝑥−5 by x−1:

1. 1. 2𝑥³÷𝑥=2𝑥²
   2. Multiply 2𝑥²(𝑥−1) =2𝑥³−2𝑥²
   3. Subtract: (2𝑥³−3𝑥²+4𝑥−5) −(2𝑥³−2𝑥²) =−𝑥²+4𝑥−5
   4. −𝑥²÷𝑥=−𝑥
   5. Multiply −𝑥(𝑥−1) =−𝑥²+𝑥
   6. Subtract:(−𝑥²+4𝑥−5) −(−𝑥²+𝑥) =3𝑥−5
   7. 3𝑥÷𝑥=3
   8. Multiply 3(𝑥−1) =3𝑥−3
   9. Subtract: (3𝑥−5) −(3𝑥−3) =−2

So, the quotient is 2𝑥²−𝑥+3with a remainder of −2.

Polynomial Operations Summary

• Addition and Subtraction: Combine like terms by adding or subtracting their coefficients.
• Multiplication: Use distributive property to multiply each term and combine like terms.
• Division: Use long division or synthetic division (for simple cases) to divide polynomials.

Understanding these operations allows you to simplify and solve polynomial equations, work with polynomial functions, and perform algebraic manipulations in calculus and beyond.