Solution - Finding the roots of polynomials

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   x² - 3x  =   x • (x - 3) 

Equation at the end of step  2  :

  ((((x2)-(3x))2)-2x•(x-3))-8

Step  3  :

Polynomial Roots Calculator :

 3.1    Find roots (zeroes) of :       F(x) = x⁴-6x³+7x²+6x-8
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  -8.

 The factor(s) are:

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

 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
   x⁴-6x³+7x²+6x-8 
can be divided by 4 different polynomials,including by  x-4 

Polynomial Long Division :

 3.2    Polynomial Long Division
Dividing :  x⁴-6x³+7x²+6x-8 
                              ("Dividend")
By         :    x-4    ("Divisor")

Quotient :  x³-2x²-x+2  Remainder:  0 

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = x³-2x²-x+2

     See theory in step 3.1
In this case, the Leading Coefficient is  1  and the Trailing Constant is  2.

 The factor(s) are:

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

 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
   x³-2x²-x+2 
can be divided by 3 different polynomials,including by  x-2 

Polynomial Long Division :

 3.4    Polynomial Long Division
Dividing :  x³-2x²-x+2 
                              ("Dividend")
By         :    x-2    ("Divisor")

Quotient :  x²-1  Remainder:  0 

Trying to factor as a Difference of Squares :

 3.5      Factoring:  x²-1 

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 : 1 is the square of 1
Check :  x²  is the square of  x¹ 

Factorization is :       (x + 1)  •  (x - 1) 

Final result :

  (x + 1) • (x - 1) • (x - 2) • (x - 4)