Topic 7.3     (Factoring by Patterns) (Special Products)

The difference of two perfect squares.

A Trinomial square

Sum or Difference of two perfect cubes.

How to determine if the expression is a perfect square.

           25x² - 121

        Is the first term a perfect square (see perfect root list)?   YES

        Is the second term a perfect square (see perfect root list)?   YES

        Is there a minus sign between the TWO terms?   YES

        Factor the problem by taking the square root of the first term plus the square

            root of the second term times the square root of the first term minus the

            square root of the second term.  

        Answer:           (5x + 11)(5x - 11)

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Samples:                x² -  16     =     (x  +  4)(x  -  4)

                                 x⁴ -  25y²  =   (x² +  5y)(x² -  5y)

                           100x² - 49y²    =   (10x  +  7y)(10x  -  7y)

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How to determine if the expression is a Trinomial square.

                            x² - 8x + 16

        Is the first term a perfect square?     YES

        Is the third term a perfect square?    YES

        Is the middle term double the product of the square root of the first and third

            term?    YES

        Factor the problem by taking the square root of the first term and the square

            root of the third term with the sign of the middle term between them

            quantity squared

            Answer        (x - 4)²

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Samples:       x² +  14x  +  49   =   (x  +  7)²

                      4x² -  12x  +    9    =   (2x  -  3)²

                        x² -   22x  +  121 =   (x  -  11)²

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How to determine if the expression is a sum or difference of two perfect cubes.

                             8x³ - 125

        Is the first term a perfect cube (see perfect root list)?   YES

        Is the second term a perfect cube (see perfect root list)?    YES

        Factor by taking the cube root of the first term (minus or plus) (whatever the

           problem had) the cube root of the second term times the square of the first

           term of the binomial you just derived opposite sign from the original binomial.

           Then take the product of the two terms you just derived and the square

           of the second term of the binomial you just derived.

        Answer:           (2x - 5)(4x² + 10x +25)

There will always be a binomial and a trinomial. (The trinomial is never factorable)

Samples:       27x³ +  1   =   (3x  +  1)(9x² -  3x  +  1)

                     125x³  -   8   =   (5x  -  2)(25x² +  10x  +  4)

                            x³  -  64  =   (x   -   4)(x²  +   4x   +   16)

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THE RULES FOR FACTORING COMPLETELY

               1.  GREATEST COMMON FACTOR

               2.  DIFFERENCE BETWEEN TWO PERFECT SQUARES

               3.  SUM OR DIFFERENCE OF TWO PERFECT CUBES

               4.  TRINOMIAL SQUARE

               5.  TRINOMIAL PRODUCT

GENERAL FACTORING         (Use steps for factoring completely above.)

     Samples:     y⁶ - 16y²    Common monomial factor?   YES   Factor this out first.

                      y²(y⁴ - 16)       Are the two terms in the parentheses a difference of two

                                                   perfect squares?     YES    Factor this next.

               y²(y² + 4)(y² - 4)    Difference of two perfect squares?     YES   Factor that.

                      y²(y² + 4)((y + 2)(y - 2)           Answer

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                49y² - 28xy + 4x²    Common monomial factor?     NO

                                                  Difference of two perfect squares?    NO

                                                  Sum or Difference of two perfect cubes?  NO

                                                  Trinomial square?   YES    Factor it.

                            (7y - 2x)²                 Answer

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               2x³ + 12x² + 18x      Common monomial factor?   YES   Factor this out first.

                  2x(x² + 6x + 9)      Difference of two perfect squares?    NO

                                                  Sum or difference of two perfect cubes?   NO

                                                  Trinomial square?    YES     Factor it.

                       2x(x + 3)²          Answer

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                     2x³ + 250            Common monomial Factor?   YES   Factor this out first.

                    2(x³ + 125)          Difference of two perfect squares?    NO

                                                  Sum or difference of two perfect cubes?   YES  Factor.

         2(x + 5)(x² - 5x + 25)     Answer

               3x² - 33x - 180          Common monomial factor?   YES   Factor this out first.

               3(x² - 11x - 60)         Difference of two perfect squares?   NO

                                                   Sum or difference of two perfect cubes?   NO

                                                   Trinomial square?   NO

                                                   Trinomial product?   YES

                 3(x + 4)(x - 15)        Answer