Question

x² – 5x – 24 x² – 64
(x+3)(x + 8) (x - 8)?
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Answer 1

Answer:

x2 - 5x - 24 Simplify ———————————— x2 - 64

Trying to factor by splitting the middle term

 1.1     Factoring  x2 - 5x - 24 

The first term is,  x2  its coefficient is  1 .

The middle term is,  -5x  its coefficient is  -5 .

The last term, "the constant", is  -24 

Step-1 : Multiply the coefficient of the first term by the constant  1 • -24 = -24 

Step-2 : Find two factors of  -24 whose sum equals the coefficient of the middle term, which is   -5 .

     -24   +   1   =   -23     -12   +   2   =   -10     -8   +   3   =   -5   That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -8  and  3 

                     x2 - 8x + 3x - 24

Step-4 : Add up the first 2 terms, pulling out like factors :

                    x • (x-8)

              Add up the last 2 terms, pulling out common factors :

                    3 • (x-8)

Step-5 : Add up the four terms of step 4 :

                    (x+3)  •  (x-8)

             Which is the desired factorization

Trying to factor as a Difference of Squares:

 1.2      Factoring:  x2-64 

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =

         A2 - AB + BA - B2 =

         A2 - AB + AB - B2 = 

         A2 - B2

Note :  AB = BA is the commutative property of multiplication. 

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 64 is the square of 8

Check :  x2  is the square of  x1 

Factorization is :       (x + 8)  •  (x - 8) 

Canceling Out :

 1.3    Cancel out  (x - 8)  which appears on both sides of the fraction line.

Final result :

x + 3 ————— x + 8