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Mastering Factoring: Difference of Two Perfect Squares

Learn how to recognize and factor the difference of two perfect squares easily with step-by-step examples and practice problems.

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Mastering Factoring: Difference of Two Perfect Squares

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  1. Algebra 1 ~ Chapter 9.5 Factoring the Difference of Two Perfect Squares

  2. 4x2–9 2x 2x3 3   A polynomial is a difference of two squares if: • There are two terms, one subtracted from the other. • Both terms are perfect squares.

  3. To Factor the difference of perfect squares, take the square root of each term. In one set of parenthesis the sign is positive, and in the other it is negative.

  4. Reading Math Recognize a difference of two squares: -the coefficients of variable terms are perfect squares -powers on variable terms are even -constants are perfect squares.

  5. 3p2– 9q4 3q2 3q2 Example 1: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 3p2– 9q4 3p2 is not a perfect square. 3p2– 9q4 is not the difference of two squares because 3p2 is not a perfect square.

  6. 100x2– 4y2 10x 10x 2y 2y   Example 2: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 100x2– 4y2 The polynomial is a difference of two squares. (10x + 2y)(10x– 2y) Write the polynomial as (a + b)(a – b). 100x2– 4y2 = (10x + 2y)(10x– 2y)

  7. x2 x2 5y3 5y3   Example 3: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. x4– 25y6 x4– 25y6 The polynomial is a difference of two squares. Write the polynomial as (a + b)(a – b). (x2 + 5y3)(x2– 5y3) x4– 25y6 = (x2 + 5y3)(x2– 5y3)

  8. 1 1 2x 2x   Check It Out! You try it… Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 1 – 4x2 1 – 4x2 (1 + 2x)(1 – 2x) 1 – 4x2 = (1+ 2x)(1 – 2x)

  9. p4 p4 7q3 7q3 ● ● Check It Out! You try another one… Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. p8– 49q6 p8– 49q6 (p4 + 7q3)(p4– 7q3) p8– 49q6 = (p4 + 7q3)(p4– 7q3)

  10. 4x 4x Check It Out! Try one more… Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 16x2– 4y5 16x2– 4y5 16x2– 4y5 is not the difference of two squares because 4y5 is not a perfect square.

  11. Example 4: Combining with GCF Factor the binomial completely. 48a3– 12a 12a(4a2– 1) 12a(2a -1)(2a +1)

  12. Example 4b: You try one… Factor the binomial completely. 3b3– 27b

  13. Example 5: Doing more than one technique Factor the binomial completely. 2x4– 162 2(x4– 81) 2(x2 -9)(x2 +9) 2(x - 3)(x + 3)(x2 +9)

  14. Example 5b: You try one… Factor the binomial completely. 4y4– 2500

  15. Example 6: Combining Factoring by grouping Factor the binomial completely. 5x3 +15x2 – 5x - 15

  16. Example 6: Your turn… Factor the binomial completely. 6x3 +30x2 – 24x - 120

  17. Example 7: Solving Equations Solve the following: p2– 25 = 0

  18. Example 7b: Solving Equations Solve the following: 18x3 = 50x

  19. Example 7c: You try one… Solve the following: 48y3 = 3y

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