Factoring polynomials is a key skill in algebra. It's all about breaking down complex expressions into simpler parts. This makes solving equations and simplifying expressions much easier.

Learning to factor helps you understand the structure of polynomials. You'll use techniques like finding common factors, grouping terms, and working with special patterns. These skills are crucial for more advanced math topics.

Factoring Polynomials

Greatest common factor identification

• Largest factor that divides all terms in an algebraic expression without a remainder
• Found by listing factors of coefficients, identifying the largest number that is a factor of all coefficients
• List variables appearing in every term, find the lowest exponent for each variable
• GCF is the product of the GCF of coefficients and common variables raised to their lowest exponents
• Essential for simplifying expressions and solving equations by factoring

Factoring out common factors

• Rewriting a polynomial as a product of factors by dividing each term by the GCF
• Steps: identify GCF of all terms, divide each term by GCF, write factored expression as product of GCF and quotient
• Example: $10x^3 + 15x^2$ factored is $5x^2(2x + 3)$ because $5x^2$ is the GCF
• Useful for simplifying complex polynomials and solving equations by factoring

Techniques for polynomial factoring

• Factoring by grouping: grouping terms, factoring out common binomial factors
  • Example: $ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)$
• Factoring trinomials using trial and error or decomposition
  • Sum/difference of cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
  • Example: $x^2 + 5x + 6 = (x + 2)(x + 3)$
• Factoring four-term polynomials by grouping or substitution
  • Example: $2x^3 + 3x^2 - 8x - 12 = (2x^2 - 8)(x + 2)$
• Applications: simplifying expressions, solving equations, finding zeros/roots, analyzing polynomial functions

Advanced factoring methods and related concepts

• Synthetic division: a shortcut method for dividing polynomials by linear factors
• Rational root theorem: helps identify potential rational roots of a polynomial equation
• Factor theorem: relates the roots of a polynomial to its factors
• Quadratic formula: used to solve quadratic equations when factoring is not possible
  • The discriminant in the quadratic formula determines the nature of the roots