To factor polynomials using the difference of squares, identify if the polynomial is in the form a² - b², and then apply the formula: a² - b² = (a + b)(a - b).
Here's a breakdown of the process:
1. Recognize the Pattern:
- The polynomial must be a binomial (two terms).
- Both terms must be perfect squares (e.g., x², 4, 9y², 16z⁴).
- The terms must be separated by a subtraction sign (hence, "difference").
2. Identify 'a' and 'b':
- Determine what expression, when squared, gives you the first term (this is 'a'). For example, if the first term is x², then a = x. If the first term is 9y², then a = 3y.
- Determine what expression, when squared, gives you the second term (this is 'b'). For example, if the second term is 4, then b = 2. If the second term is 25z⁴, then b = 5z².
3. Apply the Formula:
- Once you've identified 'a' and 'b', plug them into the formula: a² - b² = (a + b)(a - b)
Example 1:
Factor x² - 9
- a² = x² => a = x
- b² = 9 => b = 3
- Therefore, x² - 9 = (x + 3)(x - 3)
Example 2:
Factor 4y² - 25
- a² = 4y² => a = 2y
- b² = 25 => b = 5
- Therefore, 4y² - 25 = (2y + 5)(2y - 5)
Example 3:
Factor 16x⁴ - 1
- a² = 16x⁴ => a = 4x²
- b² = 1 => b = 1
- Therefore, 16x⁴ - 1 = (4x² + 1)(4x² - 1)
Important Note: In Example 3, notice that (4x² - 1) is also a difference of squares! You can factor it further:
- 4x² - 1 = (2x + 1)(2x - 1)
So, the complete factorization of 16x⁴ - 1 is: (4x² + 1)(2x + 1)(2x - 1)
Summary:
Factoring a polynomial using the difference of squares method involves recognizing the a² - b² pattern, identifying 'a' and 'b', and then applying the formula a² - b² = (a + b)(a - b). Remember to check if the resulting factors can be factored further using the difference of squares or other factoring techniques.
