Polynomial Addition and Subtraction

Polynomial addition and subtraction focus on combining like terms and writing the final expression in standard form (descending order of exponents).

POLYNOMIAL OPERATIONS
  1. Subraction & Distribution: (7x2−5x3)−(−4x2+4x3−5)=−9x3+11x2+5(7x^2 - 5x^3) - (-4x^2 + 4x^3 - 5) = -9x^3 + 11x^2 + 5
  2. Distribution with variables: (−x2+2)−(x2–1–4x)=−2x2+4x+3(−x^2 + 2) − (x^2 – 1 – 4x) = -2x^2 + 4x + 3
  3. Higher degree terms: (4+b2)−(−4b+5−5b3)=5b3+b2+4b−1(4 + b^2) - (−4b + 5 − 5b^3) = 5b^3 + b^2 + 4b - 1
  4. Direct addition: (n3−5n2)+(−3n2+7n2+5n3)=6n3−1n2(n^3 - 5n^2) + (−3n^2 + 7n^2 + 5n^3) = 6n^3 - 1n^2
  5. Multi-variable degree: (−5v3−4v4+3v)−(8v+6v2+4v2)=−4v4−5v3−10v2−5v( − 5v^3 - 4v^4 + 3v) - (8v + 6v^2 + 4v^2) = -4v^4 - 5v^3 - 10v^2 - 5v
  6. Canceling terms: (7x+x2+7)−(−x2+7x+6)=2x2+1(7x + x^2 + 7) - (−x^2 + 7x + 6) = 2x^2 + 1
  7. Fractional coefficients: (n3−n3)+115n2−14n3=−14n3+115n2(n^3 - n^3) + \frac{1}{15} n^2 - \frac{1}{4} n^3 = -\frac{1}{4} n^3 + \frac{1}{15} n^2
  8. Negative exponents & Constants: (−r3−12r)−(23+12r3)=−32r3−12r−8(−r^3 - \frac{1}{2}r) − (2^3 + \frac{1}{2}r^3) = -\frac{3}{2}r^3 - \frac{1}{2}r - 8
  9. Multi-expression simplification: (10x2+6x)−(3x+5)+(4x2−3x−17)=14x2+12(10x^2 + 6x) − (3x + 5) + (4x^2 - 3x - 17) = 14x^2 + 12
APPLICATIONS
  • Quadrilateral Perimeter: Found by summing all side lengths.
    • Example: For sides (3x−2),(2x),(2x+1),(5x−2)(3x - 2), (2x), (2x + 1), (5x - 2), the perimeter is P=12x−3P = 12x - 3.
  • Rectangle Perimeter: Uses the formula Perimeter=2(length+width)Perimeter = 2(length + width). This is used to calculate and compare pool dimensions (e.g., Olympic vs. Community pools).
CORE RULES
  • Sign Distribution: Always distribute a negative sign to every term inside the following parentheses.
  • Standard Form: Final answers must be organized by descending powers of the variable.