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Dividing polynomials is a crucial skill in algebra. It involves breaking down complex expressions into simpler parts, making it easier to solve equations and understand relationships between variables. This process is essential for tackling real-world problems in fields like engineering and physics.

Polynomial division has practical applications in areas like geometry and physics. By mastering techniques like long division and synthetic division, you'll be able to solve problems involving area, volume, and other mathematical relationships more efficiently.

Dividing Polynomials

Polynomial division methods

  • Long division of polynomials
    • Divide leading term of dividend by leading term of divisor
    • Multiply quotient by divisor and subtract result from dividend
    • Repeat process until degree of remainder is less than degree of divisor
  • Synthetic division
    • Arrange coefficients of dividend in descending order of degree
    • Write negative of constant term of divisor outside division bracket
    • Multiply first coefficient by constant term and add result to second coefficient
    • Repeat process until all coefficients have been used
    • Last number in division bracket is remainder, other numbers form quotient

Applications of polynomial division

  • Area problems
    • Divide polynomial representing area by polynomial representing one side to find other side
    • Rectangle with area $x^3 + 2x^2 - 5x - 6$ and one side $x + 3$, divide area polynomial by $x + 3$ to find other side
  • Volume problems
    • Divide polynomial representing volume by polynomial representing area of base to find height
    • Cylinder with volume $3x^3 - 7x^2 + 5x - 1$ and base area $3x - 1$, divide volume polynomial by $3x - 1$ to find height

Factors using remainder theorem

  • Remainder theorem
    • Polynomial $P(x)$ divided by $x - c$, remainder equals $P(c)$
    • If $P(c) = 0$, then $x - c$ is factor of $P(x)$
  • Finding factors using remainder theorem
    • Evaluate polynomial at various values of $x$ to find remainders
    • If remainder is 0, corresponding $x - c$ is factor of polynomial
    • To find factors of $x^3 - 2x^2 - 5x + 6$, evaluate polynomial at $x = \pm 1, \pm 2, \pm 3$
      • If $P(1) = 0$, then $x - 1$ is factor
      • If $P(-1) = 0$, then $x + 1$ is factor
      • If $P(2) = 0$, then $x - 2$ is factor

Polynomial Characteristics and Operations

  • Degree: The highest power of the variable in a polynomial
  • Leading coefficient: The coefficient of the term with the highest degree
  • Polynomial function: An expression that consists of variables and coefficients combined using addition, subtraction, and multiplication
  • Factoring: The process of breaking down a polynomial into the product of simpler polynomials
  • Rational expression: A fraction where both the numerator and denominator are polynomials