Question

What is the simplified form of the product x^2+8x+15/x-4•x^2-16/2x+6

Answer 1

Final result :

 (x + 5) • (x + 4)

 ———————

         2        

Step by step solution :

Step  1  :

           x2 - 16

Simplify   ———————

           2x + 6

Step  2  :

Pulling out like terms :

2.1     Pull out like factors :

  2x + 6  =   2 • (x + 3)

Trying to factor as a Difference of Squares :

2.2      Factoring:  x2 - 16

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 : 16 is the square of 4

Check :  x2  is the square of  x1

Factorization is :       (x + 4)  •  (x - 4)

Polynomial Long Division :

2.3    Polynomial Long Division

Dividing :  x + 4

                             ("Dividend")

By         :    x + 3    ("Divisor")

dividend     x  +  4

- divisor  * x0     x  +  3

remainder         1

Quotient :  1  

Remainder :  1  

Equation at the end of step  2  :

 (((x2)+8x)+15) (x+4)•(x-4)

 ——————————————•———————————

     (x-4)        2•(x+3)  

Step  3  :

           x2 + 8x + 15

Simplify   ————————————

              x - 4    

Trying to factor by splitting the middle term

3.1     Factoring  x2 + 8x + 15

The first term is,  x2  its coefficient is  1 .

The middle term is,  +8x  its coefficient is  8 .

The last term, "the constant", is  +15

Step-1 : Multiply the coefficient of the first term by the constant   1 • 15 = 15

Step-2 : Find two factors of  15  whose sum equals the coefficient of the middle term, which is   8 .

     -15    +    -1    =    -16

     -5    +    -3    =    -8

     -3    +    -5    =    -8

     -1    +    -15    =    -16

     1    +    15    =    16

     3    +    5    =    8    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  3  and  5

                    x2 + 3x + 5x + 15

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

                   x • (x+3)

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

                   5 • (x+3)

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

                   (x+5)  •  (x+3)

            Which is the desired factorization

Equation at the end of step  3  :

 (x + 5) • (x + 3)   (x + 4) • (x - 4)

 ————————————————— • —————————————————

       x - 4            2 • (x + 3)  

Step  4  :

Canceling Out :

4.1    Cancel out  (x+3)  which appears on both sides of the fraction line.

Canceling Out :

4.2    Cancel out  (x-4)  which appears on both sides of the fraction line.

Final result :

 (x + 5) • (x + 4)

 ——————---

         2