Polynomials Operations and Factoring

Polynomials

Addition and Subtraction

• To add or subtract polynomials, combine like terms:
  • Combine terms with the same variable(s) and coefficient(s)
  • Add or subtract coefficients, keeping the variable(s) the same
• Example: (2x^2 + 3x - 1) + (x^2 - 2x - 3) = 3x^2 + x - 4

Factoring

• Factoring is the process of expressing a polynomial as a product of simpler expressions
• Types of factoring:
  • Greatest Common Factor (GCF): factor out the largest common factor of all terms
  • Difference of Squares: a^2 - b^2 = (a + b)(a - b)
  • Sum and Difference: a^2 + 2ab + b^2 = (a + b)^2, a^2 - 2ab + b^2 = (a - b)^2
• Example: x^2 + 5x + 6 = (x + 3)(x + 2)

Multiplication

• To multiply polynomials, use the distributive property:
  • Multiply each term in one polynomial by each term in the other polynomial
  • Combine like terms
• Example: (2x + 3)(x + 4) = 2x^2 + 8x + 3x + 12 = 2x^2 + 11x + 12

Long Division

• Long division is a method for dividing a polynomial by another polynomial
• Steps:
  1. Write the dividend (divisor) and divisor
  2. Divide the first term of the dividend by the first term of the divisor
  3. Multiply the result by the divisor and subtract from the dividend
  4. Repeat steps 2-3 until the remainder is 0 or the degree of the remainder is less than the degree of the divisor
• Example: (x^2 + 3x + 2) ÷ (x + 1) = x + 2