Topic 7.2              (Factoring Polynomials II)

Factoring a trinomial product.

               (x² - 5x + 6)    =    (x - 2)(x - 3)        It makes no difference whether 2 or 3

               (x² + 5x + 6)   =    (x + 2)(x + 3)        comes first.

Determine the factors of 6 that add (the third term had a + in front of it) to get 5.

The factors of 6 are: 1, 6 & 2, 3. Use 2, 3 and each binomial will have the same sign

   as the middle term.

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               (x² - 5x - 6)  =  (x  +  1)(x  -  6)        It makes no difference which binomial

               (x² + 5x - 6) =  (x  +  6)(x  -  1)        comes first, but makes sure the largest

                                                                           factor has the sign of the middle term.

Determine the factors of 6 that subtract (the third term had a - in front of it) to

   get 5.

   The factors of 6 are: 1, 6 & 2, 3. Use 1, 6 and the largest factor will take the sign of

    the middle term

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Sample:                      x² + 7x + 12

    Factors of 12:   1, 12     &     2,  6     &     3,  4.

    Which pair of factors add (third term +) to get   7?     3, 4

    Answer:           (x + 3)(x + 4)

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Sample:                      x² - 4x - 21

   Factors of 21:      1, 21   &   3 ,7

   Which pair of factors subtract (third term -) to get 4?    3, 7

   Answer:            (x + 3)(x - 7)

Remember, largest factor of 21 takes the sign of the middle term.

Factoring trinomial products where the first term (quadratic term) has a number

    greater than 1.

                                     2x² + 11x + 5

   Find the factors of 2.    1, 2

   Find the factors of 5     1, 5

Multiply these factors together to get cross products and add (third term +)to get

   11.

   Such as:   1 x 5  +  2 x 1   =   7

   Such as:   1 x 1  +  2 x 5   =   11   OH!! That is the coefficient of the middle term.

   Place these factors so the sum of the cross products is 11.

   Answer:            (2x + 1)(x + 5)

   Remember the 5 and 2 cannot be in the same binomial.

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Factor:                       6x² + 13x - 63

   Find the factors of 6     1, 6   &   2, 3

   Find the factors of 63   1, 63   &   3, 21   &   7, 9

Multiply these factors together to get cross products and subtract (third term -)to

   get 13.

   Such as:   3 x 9   -   2 x 7  =  13   OH!! That is the coefficient of the middle term.

   Place these factors so that the difference of the cross products is 13.

   Answer:            (3x - 7)(2x + 9)

Remember the 2 and 7 cannot be in the same binomial and the 3 & 9 cannot be

 in the same binomial.  Also the produce of 3 & 9 must be positive. (+)

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