Operations with Polynomials

Adding Polynomials (x^2 + x - 6) + (x^2 + 4x + 10)
1. Step 1: Rewrite polynomials without parenthesis
  1. x^2 + x - 6 + x^2 + 4x + 10
2. Step 2: Combine like terms.
  Annotations:
  - x^2 + x^2 + x + 4x - 6 + 10=
  1. x^2 + x^2 + x + 4x - 6 + 10 = 2x^2 + 5x + 4
Multiplying Polynomials
1. For a monomial times a binomial, use the distributive property
  1. 4x(x - 2) = 4x^2 - 8x
2. For a binomial times a binomial, use FOIL (x + 2)(x + 5)
  1. (x + 2)(x + 5) = x^2 + 5x + 2x + 10
    1. Combine like terms: x^2 + 7x + 10
3. For a binomial times a trinomial (x + 3)(x^2 - 4x + 1)
  1. Step 1: Distribute the first term of the binomial to each term in the trinomial.
    1. x(x^2 - 4x + 1) = x^3 -4x^2 + 1x
  2. Step 2: Distribute the second term of the binomial to each term in the trinomial.
    1. 3(x^2 - 4x + 1) = 3x^2 -12x + 3
    2. Step 3: Combine like terms: x^3 - 4x^2 + 3x^2 + 1x - 12x + 3= x^3 - x^2 - 11x + 3
Factoring Polynomials a=1
1. Step 1: If possible, factor out the greatest common factor. Example: 3x^2 + 6x - 18=3(x^2 + 5x - 6)
  1. If it has two terms, are they both perfect squares?
    1. Yes: (a^2-b^2) Example: x^2 - 81 = (x + 9)(x - 9)
    2. No: Example: 8x - 10 = 2(4x - 5)
  2. If it has three terms, use "reverse foil"
    1. (x + p)(x + q) Example: x^2 + 10x + 16 = (x + 8)(x + 2)
    2. (x - p)(x - q) Example: x^2 - 8x + 15 = (x - 3)(x - 5)
    3. (x + p)(x - q) Example: x^2 + 5x - 14 = (x + 7)(x - 2)
Subtracting Polynomials (4x^2 -3x + 5) - (3x^2 - x - 8)
1. Step 1: Distribute the negative sign into the second polynomial.
  1. (4x^2 -3x + 5) - (-3x^2 - x - 8) = 4x^2 - 3x + 5 - 3x^2 + x + 8
2. Step 2: Combine like terms
  1. 4x^2 - 3x^2 - 3x + x + 5 - 8 = x^2 - 2x + 13

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Operations with Polynomials

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