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3.3 – Special Cases

Objectives

  • Identify and factor a difference of two squares.
  • Identify and factor a perfect square trinomial.
  • Expand binomials to produce a difference of two squares.

 

Key Terms

  • Difference of Squares – An expression that contains two perfect squares with one subtracted from the other.
    • A difference of squares can always be factored as follows: a^2-b^2 = (a + b)(a - b)
  • Difference of Two Cubes – An expression that contains two perfect cubes with one subtracted from the other.
    • A difference of two cubes can always be factored as follows: a^3-b^3=(a-b)(a^2+ab+b^2)
  • Perfect Square Trinomial – The result of multiplying a binomial by itself.
    • In other words, it is the square of a polynomial with two terms.
  • Sum of Two Cubes – An expression that contains that contains two perfect cubes, with one added to other other.
    • This type of expression can always be factored as follows: a^3+b^3=(a+b)(a^2-ab+b^2)
    • Example: 8x^3+y^6
      • 8x^3=2x\bullet 2x\bullet 2x
      • y^6=y^2\bullet y^2\bullet y^2
      • So, (2x+y^2)(4x^2-2xy^2+y^4)

 

Notes

Difference of Squares
  • Properties of the Difference of Squares
    • All variables are raised to an even power
    • There are only two terms
    • Both terms have negative coefficients
    • Both coefficients are perfect squares
    • There are only two terms
    • One term has a negative coefficient and one term has a positive coefficient
    • The middle term of the difference of squares is 0x
    • 0 times x = 0

Alg2A 3.03 - Diff Squares 02

Perfect Square Trinomials
  • Properties of Perfect Square Trinomials
    • The first and third terms must be perfect squares.
    • Neither of the perfect squares can have a minus sign
    • If the perfect square terms are A^2 and B^2 then the other term must be 2AB.

Alg2A 3.03 - PST 03

Alg2A 3.03 - Squares Plus Minus01

Alg2A 3.03 - PST 02

Difference of Two Cubes
  • Formulas and Examples

Alg2A 3.03 - Diff Cubes 01

Examples
  •  Example 1:  Difference of Squares
    • When asked to factor the expression x^2-121, a student gives the answer (x-11)(x-11).
    • What is the wrong with this answer?
    • Answer:  One of the minus signs should be a plus sign.
  • Example 2:  Difference of Squares
    • To factor 9x^2-16, you can first rewrite the expression as: (3x)^2-(4)^2.
  • Example 3:  Difference of Squares
    • What is the factorization of 16x^2 - 49?
    • Answer: (4x + 7)(4x – 7)
  • Example 4:  Difference of Squares
    • What is the factorization of 164A^2 - 81B^2?
    • Answer: (8A + 9B)(8A – 9B)
  • Example 5:  Difference of Squares
    • In the polynomial below, what number should replace the question mark to produce a difference of squares?
    • x^2 + ?x - 49
    • Answer: 0
  • Example 6:  Difference of Squares
    • What is the product of the two binomials below?
    • (4A + 5B)(4A – 5B)
    • Answer: 16A^2 - 25B^2
  • Example 7:  Sum of Two Cubes
    • Factor the expression: 343x^3 + 216y^3
    • Answer: (7x + 6y)(49x^2 - 42xy + 36y^2)
  • Example 8: Difference of Two Cubes
    • What is the factorization of 2x^2 - 16x + 32?
    • Answer: 2(x - 4)^2
  • Example 9:  Perfect Square Trinomials
    • What is the factorization of 121x^2 - 22x + 1?
    • Answer: (11x - 1)^2
  • Example 10:  Perfect Square Trinomials
    • What value in place of the question mark makes the polynomial below a perfect square trinomial? $16x^2 + ?x + 36$$
    • Answer: 48

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