Polynomial Factorization

Factorization of Polynomials

Definition

• Factorization of a polynomial is the process of expressing it as a product of simpler polynomials, called factors.

Types of Factorization

• Factorization over integers: Factoring a polynomial into factors with integer coefficients.
• Factorization over rational numbers: Factoring a polynomial into factors with rational coefficients.
• Factorization over real numbers: Factoring a polynomial into factors with real coefficients.

Methods of Factorization

• Factoring out the greatest common factor (GCF):
  • Find the GCF of all terms in the polynomial.
  • Divide each term by the GCF.
  • Write the result as a product of the GCF and the remaining polynomial.
• Factoring by grouping:
  • Group terms with common factors.
  • Factor out the common factor from each group.
  • Write the result as a product of the factors.
• Factoring quadratic polynomials:
  • Use the formula: x^2 + bx + c = (x + d)(x + e).
  • Find the values of d and e that satisfy the equation.
• Factoring by decomposition:
  • Express the polynomial as a sum of simpler polynomials.
  • Factor each simpler polynomial.
  • Write the result as a product of the factors.

Important Results

• Fundamental Theorem of Algebra: Every non-constant polynomial has at least one complex root.
• Factor Theorem: If a is a root of the polynomial f(x), then (x - a) is a factor of f(x).
• Remainder Theorem: If f(x) is divided by (x - a), the remainder is f(a).

Factorization of Polynomials

• Factorization is the process of expressing a polynomial as a product of simpler polynomials.
• Factors can be integers, rational numbers, or real numbers.

Types of Factorization

• Factorization over integers: factors have integer coefficients.
• Factorization over rational numbers: factors have rational coefficients.
• Factorization over real numbers: factors have real coefficients.

Methods of Factorization

• Factoring out the greatest common factor (GCF):
  • Find the GCF of all terms in the polynomial.
  • Divide each term by the GCF.
  • Write the result as a product of the GCF and the remaining polynomial.
• Factoring by grouping:
  • Group terms with common factors.
  • Factor out the common factor from each group.
  • Write the result as a product of the factors.
• Factoring quadratic polynomials:
  • Use the formula: x^2 + bx + c = (x + d)(x + e).
  • Find the values of d and e that satisfy the equation.
• Factoring by decomposition:
  • Express the polynomial as a sum of simpler polynomials.
  • Factor each simpler polynomial.
  • Write the result as a product of the factors.

Important Results

• Fundamental Theorem of Algebra: every non-constant polynomial has at least one complex root.
• Factor Theorem: if a is a root of the polynomial f(x), then (x - a) is a factor of f(x).
• Remainder Theorem: if f(x) is divided by (x - a), the remainder is f(a).