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Factoring The Difference of 2 SquaresAfter studying this lesson, you will be able to:
Steps of Factoring: 1. Factor out the GCF 2. Look at the number of terms:
3. Factor Completely 4. Check by Multiplying This lesson will concentrate on the second step of factoring: Factoring the Difference of 2 Squares. **When there are 2 terms, we look for the difference of 2 squares. Don't forget to look for a GCF first.** We have the difference of two squares when the following are true: There are 2 terms separated by a minus sign Each of the term is a perfect square. A partial list of perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, ... it's a good idea to memorize as many of these as you can so that you can recognize them in problems. For a variable to be a square, it must have an evennumbered exponent. For example, these are squares: x^{ 2}, x^{ 4}, a^{ 10}, x ^{2} y^{ 8}, x^{ 2} y ^{4} z^{ 2}. These, however, are not squares: x, x^{ 3}, y^{ 5}, x^{ 2}y, yz^{ 3}, a^{ 2} b^{ 5}c. To factor the difference of 2 squares, we write 2 parentheses. One will have an addition sign and the other will have a subtraction sign like this: Next, we find the square root of the first term. We put these in the first positions. Then, we find the square root of the constant term and we put these in the last positions.
Example 1 Factor x^{ 2}  64 There is no GCF other than one. So, we start with 2 parentheses. This is the difference of two squares because we have 2 terms separated by a minus sign and because each term is a perfect square. We start with the 2 parentheses and the signs. Using the sign rule for the difference of 2 squares, we put in one negative and one positive. Now we take the square root of the first term. The square root of x 2 is x so we put an x in the first positions: Now we take the square root of the constant term. The square root of 64 is 8 so we put an 8 in the last positions. Now, the problem is completely factored. (x + 8) (x  8) Check by using FOIL (x + 8) (x  8) x^{ 2} 8x + 8x  64 which is x^{ 2}  64
Example 2 Factor x^{ 2}  25 There is no GCF other than one. So, we start with 2 parentheses. This is the difference of two squares because we have 2 terms separated by a minus sign and because each term is a perfect square. We start with the 2 parentheses and the signs. Using the sign rule for the difference of 2 squares, we put in one negative and one positive. Now we take the square root of the first term. The square root of x^{ 2} is x so we put an x in the first positions: Now we take the square root of the constant term. The square root of 25 is 5 so we put a 5 in the last positions. Now, the problem is completely factored. (x + 5) (x  5) Check by using FOIL (x + 5) (x  5) x^{ 2}  5x + 5x  25 which is x^{ 2}  25
Example 3 Factor 9x^{ 2}  100 y^{ 2} There is no GCF other than one. So, we start with 2 parentheses. This is the difference of two squares because we have 2 terms separated by a minus sign and because each term is a perfect square. We start with the 2 parentheses and the signs. Using the sign rule for the difference of 2 squares, we put in one negative and one positive. Now we take the square root of the first term. The square root of 9 x 2 is 3x so we put a 3x in the first positions: Now we take the square root of the constant term. The square root of 100 y^{ 2} is 10y so we put a 10y in the last positions. Now, the problem is completely factored. (3x + 10y) (3x  10y) Check by using FOIL (3x + 10y) (3x  10y) 9x^{ 2}  30xy +30xy  100 y^{ 2 }whichis 9x^{ 2}  100 y^{ 2}
