Polynomials: Operations and Properties

Polynomials

Addition of Polynomials

• To add two or more polynomials, combine like terms
• Add corresponding coefficients
• Example: (2x^2 + 3x - 1) + (x^2 - 2x - 3) = 3x^2 + x - 4

Subtraction of Polynomials

• To subtract one polynomial from another, change the sign of the polynomial being subtracted and add
• Example: (2x^2 + 3x - 1) - (x^2 - 2x - 3) = x^2 + 5x + 2

Multiplication of Polynomials

• To multiply two polynomials, multiply each term of one polynomial by each term of the other
• Combine like terms
• Example: (2x + 3)(x + 2) = 2x^2 + 7x + 6

Factorization of Polynomials

• Factorization: expressing a polynomial as a product of simpler expressions
• Methods:
  • Factoring out greatest common factor (GCF)
  • Factoring quadratic expressions (e.g., difference of squares)
  • Factoring sum or difference of cubes
• Example: x^2 + 5x + 6 = (x + 3)(x + 2)

Remainder Theorem

• If a polynomial f(x) is divided by (x - a), the remainder is f(a)
• Example: if f(x) = x^2 + 2x - 3, then f(2) = 2^2 + 2(2) - 3 = 5, which is the remainder when dividing f(x) by (x - 2)
• The Remainder Theorem is useful for evaluating polynomials and finding roots.

Polynomials

Addition of Polynomials

• Combine like terms to add two or more polynomials
• Add corresponding coefficients to get the resulting polynomial

Subtraction of Polynomials

• Change the sign of the polynomial being subtracted and add to get the resulting polynomial
• Subtracting a polynomial is equivalent to adding its negative

Multiplication of Polynomials

• Multiply each term of one polynomial by each term of the other to get the product
• Combine like terms to simplify the resulting polynomial

Factorization of Polynomials

• Factorization is expressing a polynomial as a product of simpler expressions
• Methods of factorization include:
  • Factoring out the greatest common factor (GCF)
  • Factoring quadratic expressions (e.g., difference of squares)
  • Factoring sum or difference of cubes
• Factorization can be used to simplify polynomials and solve equations

Remainder Theorem

• The remainder of dividing a polynomial f(x) by (x - a) is f(a)
• The Remainder Theorem is useful for evaluating polynomials and finding roots
• The theorem can be used to find the remainder of dividing a polynomial by a linear factor