Question

x^3 - 12x(x - 4) - 63

Answer 1

 x3-12x(x-4)-63 

Final result :

(x2 - 9x + 21) • (x - 3)

Step by step solution :

Step  1  :

Equation at the end of step  1  :

((x3) - 12x • (x - 4)) - 63

Step  2  :

Checking for a perfect cube :

 2.1    x3-12x2+48x-63  is not a perfect cube 

Trying to factor by pulling out :

 2.2      Factoring:  x3-12x2+48x-63 

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

Group 1:  x3-63 
Group 2:  -12x2+48x 

Pull out from each group separately :

Group 1:   (x3-63) • (1)
Group 2:   (x-4) • (-12x)

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 :

 2.3    Find roots (zeroes) of :       F(x) = x3-12x2+48x-63
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  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  1 and the Trailing Constant is  -63. 

 The factor(s) are: 

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3 ,7 ,9 ,21 ,63 

 Let us test ....
  P  Q  P/Q  F(P/Q)   Divisor     -1     1      -1.00      -124.00        -3     1      -3.00      -342.00        -7     1      -7.00     -1330.00        -9     1      -9.00     -2196.00        -21     1     -21.00     -15624.00        -63     1     -63.00     -300762.00        1     1      1.00      -26.00        3     1      3.00      0.00    x-3      7     1      7.00      28.00        9     1      9.00      126.00        21     1      21.00      4914.00        63     1      63.00     205380.00   

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 
   x3-12x2+48x-63 
can be divided with  x-3 

Polynomial Long Division :

 2.4    Polynomial Long Division 
Dividing :  x3-12x2+48x-63 
                              ("Dividend")
By         :    x-3    ("Divisor")
dividend  x3 - 12x2 + 48x - 63 - divisor * x2   x3 - 3x2     remainder  - 9x2 + 48x - 63 - divisor * -9x1   - 9x2 + 27x   remainder      21x - 63 - divisor * 21x0       21x - 63 remainder       0
Quotient :  x2-9x+21  Remainder:  0 

Trying to factor by splitting the middle term

 2.5     Factoring  x2-9x+21 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -9x  its coefficient is  -9 .
The last term, "the constant", is  +21 

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

Step-2 : Find two factors of  21  whose sum equals the coefficient of the middle term, which is   -9 .
     -21   +   -1   =   -22     -7   +   -3   =   -10     -3   +   -7   =   -10     -1   +   -21   =   -22     1   +   21   =   22     3   +   7   =   10     7   +   3   =   10     21   +   1   =   22

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

Final result :

(x2 - 9x + 21) • (x - 3)

Processing ends successfully



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Answer 2

${x}^{3} - 12x(x - 4) - 63 \\ {x}^{3} - 12 {x}^{2} + 48x - 63 \\ {x}^{3} - 3 {x}^{2} - 9 {x}^{2} + 27x + 21x - 63 \\ {x}^{2} (x - 3) - 9x(x - 3) + 21(x - 3) \\ (x - 3)( {x}^{2} - 9x + 21)$