How do you determine if a polynomial is the differnce of two squares.
To determine if a polynomial is the difference of two squares, you need to look for a specific pattern within the polynomial expression. The difference of two squares is characterized by having two terms, each of which is a perfect square, with a subtraction sign (-) in between them.
Here's a step-by-step process to determine if a polynomial is the difference of two squares:
1. Identify the polynomial expression you want to analyze. For example, let's consider the expression: 𝑎^2 − 𝑏^2.
2. Look for perfect squares. Check if each term in the polynomial expression is a perfect square. A perfect square is a number or variable term that can be expressed as the square of another number or variable. For example, 𝑎^2 and 𝑏^2 are perfect squares.
3. Check for a subtraction sign. Verify that the two terms are subtracted from each other. In this case, the subtraction sign between the two perfect squares is −.
4. Apply the formula. If you have two perfect square terms separated by a subtraction sign, you can use the following formula: 𝑎^2 − 𝑏^2 = (𝑎 + 𝑏)(𝑎 − 𝑏).
5. Verify the result. Multiply the factors obtained from the formula (𝑎 + 𝑏)(𝑎 − 𝑏) and check if it equals the original polynomial expression. If it does, then the polynomial is indeed the difference of two squares.
For example, let's apply these steps to the polynomial expression 𝑥^2 − 4:
1. The polynomial expression is 𝑥^2 − 4.
2. The term 𝑥^2 is a perfect square because it can be expressed as 𝑥 * 𝑥. The term 4 is also a perfect square because it can be expressed as 2 * 2.
3. There is a subtraction sign between the two perfect squares.
4. Applying the formula, we get: 𝑥^2 − 4 = (𝑥 + 2)(𝑥 − 2).
5. To verify, we multiply the factors: (𝑥 + 2)(𝑥 − 2) = 𝑥 * 𝑥 − 2 * 𝑥 + 2 * 𝑥 - 2 * 2 = 𝑥^2 − 4.
Since the product of the factors matches the original polynomial expression, we can conclude that 𝑥^2 − 4 is the difference of two squares.