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Factoring Special Forms: Difference of Two Squares and Perfect Square Trinomials

Learn how to factor expressions using the difference of two squares and perfect square trinomials methods with the help of area models and rules.

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Factoring Special Forms: Difference of Two Squares and Perfect Square Trinomials

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  1. Chapter 5 Section 5 • Factoring Special Forms

  2. Factoring Difference of Two Squares • Two ways • Area model • Rule

  3. Factor 9x2 - 100 Area model: set up 9x2 – 100 9x2 + 0x – 100 Sum is 0, product is - 900 Two numbers are 30 and - 30

  4. Factor 9x2 - 100 Rewrite as the difference of two squares 9x2 is (3x)2 100 is 102 So (3x)2 - 102 Rule: Factor the difference of two square as the product of the sum and difference of the those terms. Thus: (3x – 10)(3x+ 10)

  5. Caution • Be sure that you factor the common factor before factoring

  6. Try • 4x2 – 9 • 2x3 – 8x

  7. Repeated Factorization • Try: 81x4 – 16 • Note: After you factor, look at the factors.

  8. Factoring Perfect Square Trinomials • Two ways • Area Model • Rule

  9. Check for a Perfect Square Trinomial • Middle (linear) term is twice the product of the outside terms that are being squared. • Example: 4x2 + 12xy + 9y2 • leading term: (2x)2 • Last term: (3y)2 • Multiply the expression being square and multiply by 2 • 2(2x)(3y) which is the middle term.

  10. Factor: 4x2 + 12xy + 9y2 • Since it is a perfect square trinomial • (2x)2 + 2(2x)(3y) + (3y)2 • Factor: Take the terms being squared in order and write as a binomial with the first sign and square the binomial. • (2x + 3y)2

  11. Factor: 4x2 + 12xy + 9y2 • Set up for the area model • Sum is 12, product is (4)(9) • Numbers are 6xy and 6xy

  12. Sum or Difference of Two Cube • Observe: • (A + B)3 = (A + B)(A2 - AB + B2) • (A - B)3 = (A - B)(A2 + AB + B2) • Words: Write the binomial, then square the first, change the signs, multiply together and square the second.

  13. Factor: x3 + 125 • Rewrite as the sum of cubes • (x)3 + (5)3

  14. Try • 25x4 – 25y6 • 2x3y – 18xy • 25y2 – 10y + 1 • 27x3 - 8

  15. Summary • Binomial • Difference of two squares • Sum or Difference of two cubes • Trinomial • Area model • Grouping • Perfect square trinomial

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