Difference of Squares Formula (advanced version)

Table of Contents

What is the Difference of Squares Formula?

The difference of squares formula is an algebraic identity that expresses the difference between two squares as the product of their sum and difference. It is represented mathematically by the following equation:

a^2 - b^2 = (a + b)(a - b)

where a and b can be any real numbers or algebraic expressions.

How Does the Difference of Squares Formula Work?

The difference of squares formula is based on the distributive property of multiplication. Let's break down the formula step-by-step:

  1. We start with the expression a^2 - b^2.
  2. We can rewrite a^2 as (a)(a) and b^2 as (b)(b).
  3. Using the distributive property, we can expand the expression as ({a} \times {a}) - ({b} \times {b}).
  4. The distributive property states that a(b + c) = ab + ac. In this case, we can rewrite the expression as ({a} \times 1 \times {a}) - ({b} \times 1 \times {b}).
  5. Simplifying the expression, we get (a + b)(a - b).

Therefore, we have proven that a^2 - b^2 = (a + b)(a - b).

Four Examples of Using the Difference of Squares Formula

Example 1: Factoring a Simple Polynomial

Let's factor the polynomial x^2 - 9.

  1. Step 1: Identify the terms of the polynomial. In this case, we have x^2 and -9. Notice that x^2 is of the form a^2 where a = x and -9 is a perfect square since it is equal to to $3^2$.
  2. Step 2: Apply the difference of squares formula. Since we have identified a perfect square term ($x^2$) and a constant term that is also a perfect square ($-9$), we can rewrite the polynomial as $x^2 - 3^2$. Now, we can directly apply the difference of squares formula. Thus, the factored form of the polynomial is $(x + 3)(x - 3)$.

Example 2: Factoring a Polynomial with a Negative Coefficient

Conclusion

The difference of squares formula is a powerful tool for factoring polynomial expressions. By understanding the formula and its applications, you can simplify complex expressions and solve equations more efficiently. Remember to practice with various polynomial expressions to solidify your grasp of this concept. There are many online resources and textbooks that provide additional practice problems.