Question

multiply : x^2 + 2/3x + 5/6x by x-7/4

Answer 1

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Rewrite the whole as a fraction using  3  as the denominator :

x2 x2 • 3 x2 = —— = —————— 1 3
Adding fractions that have a common denominator :

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

x2 • 3 + 2x 3x2 + 2x ——————————— = ———————— 3 3
Equation at the end of step  4  : (3x2 + 2x) 5x2 7 (—————————— + ———) - — 3 6 4 Step  5  :Step  6  :Pulling out like terms :

 6.1     Pull out like factors :

   3x2 + 2x  =   x • (3x + 2) 

Calculating the Least Common Multiple :

 6.2    Find the Least Common Multiple 

      The left denominator is :       3 

      The right denominator is :       6 

Calculating Multipliers :

 6.3    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 = 2

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 6.4      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. x • (3x+2) • 2 —————————————————— = —————————————— L.C.M 6 R. Mult. • R. Num. 5x2 —————————————————— = ——— L.C.M 6 Adding fractions that have a common denominator :

 6.5       Adding up the two equivalent fractions 

x • (3x+2) • 2 + 5x2 11x2 + 4x ———————————————————— = ————————— 6 6 Equation at the end of step  6  : (11x2 + 4x) 7 ——————————— - — 6 4 Step  7  :Step  8  :Pulling out like terms :

 8.1     Pull out like factors :

   11x2 + 4x  =   x • (11x + 4) 

Calculating the Least Common Multiple :

 8.2    Find the Least Common Multiple 

      The left denominator is :       6 

      The right denominator is :       4 

Calculating Multipliers :

 8.3    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 = 2

   Right_M = L.C.M / R_Deno = 3

Making Equivalent Fractions :

 8.4      Rewrite the two fractions into equivalent fractions

L. Mult. • L. Num. x • (11x+4) • 2 —————————————————— = ——————————————— L.C.M 12 R. Mult. • R. Num. 7 • 3 —————————————————— = ————— L.C.M 12 Adding fractions that have a common denominator :

 8.5       Adding up the two equivalent fractions 

x • (11x+4) • 2 - (7 • 3) 22x2 + 8x - 21 ————————————————————————— = —————————————— 12 12 Trying to factor by splitting the middle term

 8.6     Factoring  22x2 + 8x - 21 

The first term is,  22x2  its coefficient is  22 .
The middle term is,  +8x  its coefficient is  8 .
The last term, "the constant", is  -21 

Step-1 : Multiply the coefficient of the first term by the constant   22 • -21 = -462 

Step-2 : Find two factors of  -462  whose sum equals the coefficient of the middle term, which is   8 .

     -462   +   1   =   -461     -231   +   2   =   -229     -154   +   3   =   -151     -77   +   6   =   -71     -66   +   7   =   -59     -42   +   11   =   -31     -33   +   14   =   -19     -22   +   21   =   -1     -21   +   22   =   1     -14   +   33   =   19     -11   +   42   =   31     -7   +   66   =   59     -6   +   77   =   71     -3   +   154   =   151     -2   +   231   =   229     -1   +   462   =   461

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

Final result : 22x2 + 8x - 21 —————————————— 12
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