Factoring is the reverse process of expanding binomial expressions. It's a crucial skill for simplifying expressions and, most importantly, for solving quadratic equations later on. This lesson covers how to break down trinomial quadratic expressions into their original binomial factors.

1. Core Concepts (Short Notes)

• Trinomial Quadratic Expression: An expression with three terms in the form ax² + bx + c, where 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term.

• Factoring: The process of finding the two binomial expressions that, when multiplied together, give you the original quadratic expression. You're essentially finding the original "ingredients" of the expression.

• Difference of Two Squares: A special type of binomial expression (a² - b²), which does not look like a trinomial but has a very specific and easy-to-remember pair of factors.

2. Key Methods & Formulas

Method 1: Factoring x² + bx + c (when a = 1)

This is the most common type you'll see. The "Product-Sum" Method: Look for two numbers that:

1. Multiply to give the constant term (c).

2. Add to give the coefficient of x (b).

Example: Factor x² + 7x + 10

• We need two numbers that multiply to +10 and add to +7.

• Factors of 10: (1, 10), (2, 5).

• Let's check their sums: 1+10=11, 2+5=7. The correct pair is 2 and 5.

• Answer: (x + 2)(x + 5)

Example with Negatives: Factor x² - 8a + 12

• We need two numbers that multiply to +12 and add to -8.

• Since the product is positive but the sum is negative, both numbers must be negative.

• Factors of 12: (-1, -12), (-2, -6), (-3, -4).

• Check sums: -1+(-12)=-13, -2+(-6)=-8. The correct pair is -2 and -6.

• Answer: (a - 2)(a - 6)

Method 2: Factoring ax² + bx + c (when a ≠ 1)

This is a slightly longer process called Factoring by Grouping.

1. Find two numbers that multiply to a × c and add to b.

2. Use these two numbers to split the middle term (bx) into two separate terms.

3. Factor the resulting four-term expression by grouping the first two terms and the last two terms.

Example: Factor 3x² + 14x + 15

1. Product-Sum: We need numbers that multiply to a × c = 3 × 15 = 45 and add to b = 14. The numbers are 9 and 5.

2. Split Middle Term: Rewrite 14x as 9x + 5x. The expression becomes 3x² + 9x + 5x + 15.

3. Factor by Grouping:

   • Group the first two terms: 3x² + 9x = 3x(x + 3)

   • Group the last two terms: 5x + 15 = 5(x + 3)

   • The expression is now 3x(x + 3) + 5(x + 3).

4. Final Factorization: The common bracket is (x + 3). The terms outside the brackets form the other factor: (3x + 5).

   • Answer: (x + 3)(3x + 5)

Formula 1: Difference of Two Squares

This is a vital shortcut you must recognize immediately. Formula: a² - b² = (a - b)(a + b)

Example: Factor x² - 36

• Rewrite as (x)² - (6)². Here, a = x and b = 6.

• Answer: (x - 6)(x + 6)

Example: Factor 25a² - 16b²

• Rewrite as (5a)² - (4b)². Here, a = 5a and b = 4b.

• Answer: (5a - 4b)(5a + 4b)

3. Tips & Tricks for Exams

• Always Check for a Common Factor First! Before trying any other method, see if you can factor out a number or variable from all terms.

  • Example: 2x² - 10x + 12. First, factor out the 2: 2(x² - 5x + 6). Now factor the simpler trinomial inside: 2(x - 2)(x - 3).

• Check Your Answer: After factoring, you can quickly multiply your answer back out. If you get the original expression, you know you are correct.

• Signs are Critical: Pay close attention to the signs of b and c.

  • If c is positive, the two numbers have the same sign (both + or both -).

  • If c is negative, the two numbers have different signs (one +, one -).

4. Important Points to Remember

• Factoring is the opposite of expanding.

• The goal is to turn a sum of terms into a product of factors (brackets).

• Not all quadratic expressions can be factored using simple integers.

• Mastering factoring is essential for solving quadratic equations, which is a major part of your algebra syllabus.