Solution - Adding, subtracting and finding the least common multiple

Step  1  :

            a2 + 3a + 9
 Simplify   ———————————
               a - 2   

Trying to factor by splitting the middle term

 1.1     Factoring  a² + 3a + 9 

The first term is,  a²  its coefficient is  1 .
The middle term is,  +3a  its coefficient is  3 .
The last term, "the constant", is  +9 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 9 = 9 

Step-2 : Find two factors of  9  whose sum equals the coefficient of the middle term, which is   3 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Polynomial Long Division :

 1.2    Polynomial Long Division
Dividing :  a²+3a+9 
                              ("Dividend")
By         :    a-2    ("Divisor")

Quotient :  a+5 
Remainder :  19 

Equation at the end of step  1  :

  ((a3)-27) (a2+3a+9)
  —————————+—————————
  ((a2)-4)     a-2   

Step  2  :

            a3 - 27
 Simplify   ———————
            a2 - 4 

Trying to factor as a Difference of Cubes:

 2.1      Factoring:  a³ - 27 

Theory : A difference of two perfect cubes,  a³ - b³ can be factored into
              (a-b) • (a² +ab +b²)

Proof :  (a-b)•(a²+ab+b²) =
            a³+a²b+ab²-ba²-b²a-b³ =
            a³+(a²b-ba²)+(ab²-b²a)-b³ =
            a³+0+0-b³ =
            a³-b³

Check :  27  is the cube of   3 
Check :  a³ is the cube of   a¹

Factorization is :
             (a - 3)  •  (a² + 3a + 9) 

Trying to factor by splitting the middle term

 2.2     Factoring  a² + 3a + 9 

The first term is,  a²  its coefficient is  1 .
The middle term is,  +3a  its coefficient is  3 .
The last term, "the constant", is  +9 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 9 = 9 

Step-2 : Find two factors of  9  whose sum equals the coefficient of the middle term, which is   3 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Trying to factor as a Difference of Squares :

 2.3      Factoring:  a²-4 

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 : 4 is the square of 2
Check :  a²  is the square of  a¹ 

Factorization is :       (a + 2)  •  (a - 2) 

Polynomial Long Division :

 2.4    Polynomial Long Division
Dividing :  a - 3 
                              ("Dividend")
By         :    a + 2    ("Divisor")

Quotient :  1 
Remainder :  -5 

Equation at the end of step  2  :

  (a - 3) • (a2 + 3a + 9)    (a2 + 3a + 9)
  ——————————————————————— +  —————————————
     (a + 2) • (a - 2)           a - 2    

Step  3  :

Calculating the Least Common Multiple :

 3.1    Find the Least Common Multiple

      The left denominator is :       (a+2) • (a-2) 

      The right denominator is :       a-2 

      Least Common Multiple:
      (a+2) • (a-2) 

Calculating Multipliers :

 3.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 1

   Right_M = L.C.M / R_Deno = a+2

Making Equivalent Fractions :

 3.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.      (a-3) • (a2+3a+9)
   ——————————————————  =   —————————————————
         L.C.M               (a+2) • (a-2)  

   R. Mult. • R. Num.      (a2+3a+9) • (a+2)
   ——————————————————  =   —————————————————
         L.C.M               (a+2) • (a-2)  

Adding fractions that have a common denominator :

 3.4       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 (a-3) • (a2+3a+9) + (a2+3a+9) • (a+2)     2a3 + 5a2 + 15a - 9
 —————————————————————————————————————  =  ———————————————————
             (a+2) • (a-2)                  (a + 2) • (a - 2) 

Checking for a perfect cube :

 3.5    2a³ + 5a² + 15a - 9  is not a perfect cube

Trying to factor by pulling out :

 3.6      Factoring:  2a³ + 5a² + 15a - 9 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  15a - 9 
Group 2:  5a² + 2a³ 

Pull out from each group separately :

Group 1:   (5a - 3) • (3)
Group 2:   (2a + 5) • (a²)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

 3.7    Find roots (zeroes) of :       F(a) = 2a³ + 5a² + 15a - 9
Polynomial Roots Calculator is a set of methods aimed at finding values of  a  for which   F(a)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  a  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  2  and the Trailing Constant is  -9.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1 ,3 ,9

 Let us test ....

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
   2a³ + 5a² + 15a - 9 
can be divided with  2a - 1 

Polynomial Long Division :

 3.8    Polynomial Long Division
Dividing :  2a³ + 5a² + 15a - 9 
                              ("Dividend")
By         :    2a - 1    ("Divisor")

Quotient :  a²+3a+9  Remainder:  0 

Trying to factor by splitting the middle term

 3.9     Factoring  a²+3a+9 

The first term is,  a²  its coefficient is  1 .
The middle term is,  +3a  its coefficient is  3 .
The last term, "the constant", is  +9 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 9 = 9 

Step-2 : Find two factors of  9  whose sum equals the coefficient of the middle term, which is   3 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  (a2 + 3a + 9) • (2a + 1)
  ————————————————————————
     (a + 2) • (a + 2)