🔟Elementary Algebra Unit 7 – Factoring

What's Factoring All About?

• Factoring involves breaking down a polynomial expression into smaller expressions that can be multiplied together to get the original expression
• Factoring is the opposite process of expanding or multiplying out expressions
• The factors of a number are the numbers that divide evenly into it with no remainder (12 has factors of 1, 2, 3, 4, 6, and 12)
• In algebra, factoring involves finding the factors of a polynomial expression
  • These factors are usually written in parentheses and multiplied together
• Factoring is an essential skill for simplifying expressions, solving equations, and working with quadratic functions
• Understanding the concepts of greatest common factor (GCF) and grouping is crucial for successful factoring
• Factoring can help reveal the zeros or roots of a polynomial function, which are the x-values where the function equals zero

Types of Factoring You'll See

• Common factor factoring: involves finding the greatest common factor (GCF) among all terms and factoring it out
  • Example: $6x^2 + 9x = 3x(2x + 3)$, where $3x$ is the GCF
• Grouping: involves grouping terms together and factoring out common binomials
  • Example: $ax + ay + bx + by = (ax + bx) + (ay + by) = x(a + b) + y(a + b) = (a + b)(x + y)$
• Difference of squares: a special form of factoring where a perfect square minus another perfect square is factored into two binomials
  • Example: $x^2 - 9 = (x + 3)(x - 3)$
• Trinomial factoring: factoring a quadratic expression in the form $ax^2 + bx + c$ into two binomials
  • Example: $x^2 + 5x + 6 = (x + 2)(x + 3)$
• Sum or difference of cubes: special forms of factoring where the sum or difference of two perfect cubes is factored into binomials
  • Example: $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$
• Factoring by substitution: involves substituting a variable to simplify the expression before factoring
• Factoring with multiple variables: involves factoring expressions with more than one variable, using the same techniques as single-variable expressions

Key Terms to Know

• Factor: a number or expression that divides evenly into another number or expression
• Factoring: the process of breaking down a polynomial expression into smaller expressions that can be multiplied together to get the original expression
• Polynomial: an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents
• Monomial: a polynomial with only one term (e.g., $3x^2$)
• Binomial: a polynomial with two terms (e.g., $x^2 + 2x$)
• Trinomial: a polynomial with three terms (e.g., $x^2 + 2x + 1$)
• Coefficient: a constant multiplied by a variable in a term (e.g., in the term $4x$, the coefficient is 4)
• Greatest common factor (GCF): the largest factor that divides evenly into each term of a polynomial
• Quadratic expression: a polynomial where the highest exponent of the variable is 2 (e.g., $ax^2 + bx + c$)
• Perfect square trinomial: a trinomial that can be factored into two identical binomials (e.g., $x^2 + 6x + 9 = (x + 3)^2$)

Step-by-Step Factoring Techniques

1. Identify the type of polynomial expression you are working with (monomial, binomial, trinomial, etc.)
2. Look for the greatest common factor (GCF) among all terms
   • Factor out the GCF if one exists
3. If the expression is a binomial, check for special forms like the difference of squares or the sum or difference of cubes
   • If it matches a special form, factor according to the corresponding formula
4. If the expression is a trinomial, follow these steps: a. Multiply the coefficient of the $x^2$ term (a) by the constant term (c) b. Find the factors of the product from step a that add up to the coefficient of the x term (b) c. Rewrite the trinomial using the factors from step b and factor by grouping
5. If the expression has four or more terms, try factoring by grouping
   • Group the terms into pairs and factor out any common factors
   • Repeat the process until a common binomial emerges, then factor it out
6. If the expression involves multiple variables, use the same techniques as single-variable expressions, treating each variable separately
7. Always check your answer by multiplying the factored expression to ensure it equals the original expression

Common Mistakes and How to Avoid Them

• Forgetting to look for the greatest common factor (GCF) before attempting other factoring methods
  • Always start by checking for a GCF and factor it out first
• Incorrectly identifying the type of polynomial expression
  • Take a moment to count the number of terms and identify any special forms
• Misapplying the difference of squares or sum/difference of cubes formulas
  • Double-check that the expression perfectly matches the required form before applying these formulas
• Making arithmetic errors when finding factors or multiplying expressions
  • Take your time and double-check your calculations
• Overlooking negative signs or coefficients
  • Pay close attention to the signs of each term and factor
• Attempting to factor expressions that are not factorable
  • Not all polynomial expressions can be factored; some are prime or irreducible
• Forgetting to check your answer by multiplying the factored expression
  • Always verify your work to ensure the factored expression is equivalent to the original

Real-World Applications

• Factoring is used in solving quadratic equations, which have applications in physics, engineering, and finance
  • Example: the trajectory of a projectile can be modeled using a quadratic equation
• Factoring helps simplify complex fractions and rational expressions, which are used in various fields like chemistry and economics
• In computer science, factoring is used in cryptography and prime number generation
  • Example: the RSA encryption algorithm relies on the difficulty of factoring large numbers
• Factoring is essential for working with polynomials, which are used to model real-world phenomena such as population growth and the motion of objects
• Understanding factoring can help in analyzing the behavior of quadratic functions and their graphs, which have applications in business and science
  • Example: the profit of a company can be modeled as a quadratic function of the number of units produced
• Factoring is a fundamental skill in algebra and is necessary for more advanced mathematical concepts like calculus and linear algebra

Practice Problems and Solutions

1. Factor $12x^2 - 27x$ Solution: $12x^2 - 27x = 3x(4x - 9)$

2. Factor $x^2 - 64$ Solution: $x^2 - 64 = (x + 8)(x - 8)$

3. Factor $x^3 - 27$ Solution: $x^3 - 27 = (x - 3)(x^2 + 3x + 9)$

4. Factor $2x^2 + 7x + 3$ Solution: $2x^2 + 7x + 3 = (2x + 1)(x + 3)$

5. Factor $6x^2 + 5x - 6$ Solution: $6x^2 + 5x - 6 = (3x - 2)(2x + 3)$

6. Factor $4x^2 - 9y^2$ Solution: $4x^2 - 9y^2 = (2x + 3y)(2x - 3y)$

7. Factor $3x^3 + 5x^2 - 2x$ Solution: $3x^3 + 5x^2 - 2x = x(3x^2 + 5x - 2) = x(3x - 1)(x + 2)$

Tips for Mastering Factoring

• Practice regularly to develop a strong understanding of the different factoring techniques and when to use them
• Break down the factoring process into smaller steps and focus on one step at a time
• Look for patterns and common forms like the difference of squares or perfect square trinomials
• Use the reverse FOIL method (First, Outer, Inner, Last) to help factor trinomials
  • Example: to factor $x^2 + 5x + 6$, think of two numbers that multiply to 6 and add to 5 (2 and 3)
• When factoring expressions with multiple variables, treat each variable separately and look for common factors
• Utilize online resources, such as factoring calculators and practice problems, to check your work and gain additional practice
• Understand the connection between factoring and multiplying polynomials; practicing multiplication can reinforce your factoring skills
• Remember that factoring is a skill that takes time and practice to master; don't get discouraged if it seems challenging at first