📘Intermediate Algebra Unit 6 – Factoring

What's Factoring All About?

• Factoring breaks down an expression into its component parts multiplied together
• Reverses the process of multiplication to find the factors that produce a given expression
• Helps simplify complex expressions into a product of simpler terms
• Useful for solving equations, simplifying fractions, and finding roots of polynomials
• Factored form reveals important features of the original expression
  • Zeros of the expression correspond to the roots of the factored terms
  • Degree of the polynomial is the sum of the degrees of the factored terms
• Factoring is a fundamental skill in algebra with applications in higher mathematics (calculus, linear algebra)
• Mastering factoring techniques enhances problem-solving abilities and mathematical reasoning

Types of Factoring

• Common factor factoring extracts the greatest common factor (GCF) from all terms
• Grouping method involves arranging terms into groups with a common factor, then factoring out the GCF
• Difference of squares factoring applies to expressions in the form $a^2 - b^2$, resulting in $(a+b)(a-b)$
• Perfect square trinomials have the form $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$, factoring into $(a+b)^2$ or $(a-b)^2$
• Sum or difference of cubes factoring applies to $a^3 + b^3$ or $a^3 - b^3$
  • $a^3 + b^3 = (a+b)(a^2-ab+b^2)$
  • $a^3 - b^3 = (a-b)(a^2+ab+b^2)$
• Quadratic expressions $ax^2 + bx + c$ can be factored using various methods (trial and error, decomposition, quadratic formula)
• Choosing the appropriate factoring method depends on the structure of the given expression

Common Factor Method

• Identify the greatest common factor (GCF) among all terms in the expression
• Factor out the GCF, leaving the remaining factors in parentheses
• GCF can include both numerical coefficients and variable factors
  • Example: $6x^2 + 9x = 3x(2x + 3)$, where $3x$ is the GCF
• If a negative factor is extracted, the sign of the remaining terms in parentheses may change
• Factoring out the GCF is the first step in most factoring problems
• Helps simplify expressions and makes further factoring steps easier
• Can be combined with other factoring methods for more complex expressions

Grouping Method

• Arrange the terms of the expression into two or more groups with a common factor
• Factor out the common factor from each group
• If the remaining factors in parentheses are the same, factor out the common binomial
  • Example: $ax + ay + bx + by = (ax + bx) + (ay + by) = x(a + b) + y(a + b) = (a + b)(x + y)$
• Grouping is useful when the expression has four or more terms
• Helps break down the problem into smaller, more manageable parts
• Requires careful arrangement of terms to create groups with common factors
• May involve multiple steps of grouping and factoring to completely factor the expression

Special Factoring Patterns

• Difference of squares: $a^2 - b^2 = (a+b)(a-b)$
  • Example: $x^2 - 9 = (x+3)(x-3)$
• Perfect square trinomials: $a^2 + 2ab + b^2 = (a+b)^2$ and $a^2 - 2ab + b^2 = (a-b)^2$
  • Example: $x^2 + 6x + 9 = (x+3)^2$
• Sum of cubes: $a^3 + b^3 = (a+b)(a^2-ab+b^2)$
  • Example: $x^3 + 8 = (x+2)(x^2-2x+4)$
• Difference of cubes: $a^3 - b^3 = (a-b)(a^2+ab+b^2)$
  • Example: $x^3 - 27 = (x-3)(x^2+3x+9)$
• Recognizing these patterns allows for quick factoring without trial and error
• Memorizing the formulas for these special cases saves time and effort
• These patterns frequently appear in more complex factoring problems

Factoring Quadratic Expressions

• Quadratic expressions have the general form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$
• Goal is to factor the quadratic into the product of two linear terms: $(px + q)(rx + s)$
• Trial and error method involves guessing factors based on the product $ac$ and sum $b$
• Decomposition method splits the middle term $bx$ into two terms with a common factor
  • Example: $x^2 + 5x + 6 = x^2 + 2x + 3x + 6 = x(x + 2) + 3(x + 2) = (x + 3)(x + 2)$
• Quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ can be used to find the roots, which can then be used to factor the expression
• Factoring quadratics is essential for solving quadratic equations and finding zeros of functions
• Factored form reveals the x-intercepts of the corresponding parabola

Solving Equations Using Factoring

• Set the factored expression equal to zero
• Apply the zero product property: if $ab = 0$, then either $a = 0$ or $b = 0$ (or both)
• Solve each linear factor separately to find the solution(s)
  • Example: $(x-2)(x+3) = 0$ implies $x-2 = 0$ or $x+3 = 0$, so $x = 2$ or $x = -3$
• Factoring is an efficient method for solving quadratic equations
• Solutions correspond to the x-intercepts of the related quadratic function
• Checking solutions by substituting them back into the original equation verifies the factorization
• Factoring can also be used to solve higher-degree polynomial equations
• Some equations may have no real solutions if the factors cannot be set equal to zero

Real-World Applications

• Factoring is used in various fields, including physics, engineering, and economics
• Helps simplify complex equations that model real-world phenomena
• Used in optimization problems to find minimum or maximum values (cost, profit, area)
  • Example: minimizing material cost for a box with a given volume
• Factors can represent important properties or components in a system
• Solving factored equations can determine critical points, equilibrium states, or breakeven points
• Factoring is a key tool in analyzing quadratic functions that model projectile motion, free fall, or other physical scenarios
• Factoring techniques are essential for advanced mathematics courses and problem-solving in real-world contexts