Solution - Finding the roots of polynomials

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (((2 • (x3)) -  24x2) -  8x) +  64  = 0 
 Step  2  :

Equation at the end of step  2  :

  ((2x3 -  24x2) -  8x) +  64  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   2x³ - 16x² - 8x + 64  = 

  2 • (x³ - 8x² - 4x + 32) 

Checking for a perfect cube :

 4.2    x³ - 8x² - 4x + 32  is not a perfect cube

Trying to factor by pulling out :

 4.3      Factoring:  x³ - 8x² - 4x + 32 

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

Group 1:  x³ + 32 
Group 2:  -8x² - 4x 

Pull out from each group separately :

Group 1:   (x³ + 32) • (1)
Group 2:   (2x + 1) • (-4x)

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 :

 4.4    Find roots (zeroes) of :       F(x) = x³ - 8x² - 4x + 32
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  32.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16 ,32

 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
   x³ - 8x² - 4x + 32 
can be divided by 3 different polynomials,including by  x - 8 

Polynomial Long Division :

 4.5    Polynomial Long Division
Dividing :  x³ - 8x² - 4x + 32 
                              ("Dividend")
By         :    x - 8    ("Divisor")

Quotient :  x²-4  Remainder:  0 

Trying to factor as a Difference of Squares :

 4.6      Factoring:  x²-4 

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 :  x²  is the square of  x¹ 

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

Equation at the end of step  4  :

  2 • (x + 2) • (x - 2) • (x - 8)  = 0 

Step  5  :

Theory - Roots of a product :

 5.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Equations which are never true :

 5.2      Solve :    2   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 5.3      Solve  :    x+2 = 0 

 Subtract  2  from both sides of the equation : 
                      x = -2

Solving a Single Variable Equation :

 5.4      Solve  :    x-2 = 0 

 Add  2  to both sides of the equation : 
                      x = 2

Solving a Single Variable Equation :

 5.5      Solve  :    x-8 = 0 

 Add  8  to both sides of the equation : 
                      x = 8

Three solutions were found :

1.  x = 8
2.  x = 2
3.  x = -2