Factoring polynomials with the difference of squares pattern is a straightforward process that relies on recognizing a specific structure.
Understanding the Difference of Squares Pattern
The difference of squares pattern states that the difference of two perfect squares can be factored into the product of the sum and difference of their square roots. This is represented by the following formula:
a² - b² = (a + b)(a - b)
Where:
- a² and b² are perfect squares.
- a and b are the square roots of a² and b², respectively.
Steps to Factor Polynomials with Difference of Squares
- Identify the perfect squares: Look for terms in the polynomial that are perfect squares. A perfect square is a number that can be obtained by squaring another number. For example, 9 is a perfect square because it is the square of 3 (3² = 9).
- Recognize the difference: Ensure that the two perfect squares are being subtracted from each other.
- Apply the formula: Use the formula a² - b² = (a + b)(a - b) to factor the polynomial.
- Simplify: If possible, simplify the factored expression.
Examples
Example 1:
Factor the polynomial x² - 25.
- Identify the perfect squares: x² is the square of x, and 25 is the square of 5.
- Recognize the difference: The terms are being subtracted.
- Apply the formula: x² - 25 = (x + 5)(x - 5)
Example 2:
Factor the polynomial 4y² - 9.
- Identify the perfect squares: 4y² is the square of 2y, and 9 is the square of 3.
- Recognize the difference: The terms are being subtracted.
- Apply the formula: 4y² - 9 = (2y + 3)(2y - 3)
Practical Insights
- Recognizing the pattern: Practice identifying perfect squares and the difference of squares pattern to make factoring easier.
- Applying the formula: Memorize the formula a² - b² = (a + b)(a - b) to apply it effectively.
- Simplifying: Always simplify the factored expression if possible.
