Simplify the rational expression t^2=2t-24/t^2-36 and state any restrictions on the variable

t - 4

 —————

 t - 6

Step-by-step explanation:

           t2 + 2t - 24

Simplify   ————————————

             t2 - 36    

Trying to factor by splitting the middle term

   Factoring  t2 + 2t - 24  

The first term is,  t2  its coefficient is  1 .

The middle term is,  +2t  its coefficient is  2 .

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

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

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

     -24    +    1    =    -23  

     -12    +    2    =    -10  

     -8    +    3    =    -5  

     -6    +    4    =    -2  

     -4    +    6    =    2    That's it

Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -4  and  6  

                    t2 - 4t + 6t - 24

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

                   t • (t-4)

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

                   6 • (t-4)

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

                   (t+6)  •  (t-4)

            Which is the desired factorization

Trying to factor as a Difference of Squares :

1.2      Factoring:  t2-36  

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 : 36 is the square of 6

Check :  t2  is the square of  t1  

Factorization is :       (t + 6)  •  (t - 6)  

Canceling Out :

   Cancel out  (t + 6)  which appears on both sides of the fraction line.

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step-by-step explanation:

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