Solution - Factoring multivariable polynomials

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

      x^2*(y-z)+y^2*(z-x)+z^2*(x-y)-((x-y)*(z-y)*(y-x))=0 

Step  1  :

Equation at the end of step  1  :

  ((((x2)•(y-z))+((y2)•(z-x)))+((z2)•(x-y)))-((x-y)•(z-y)•(y-x))  = 0 

Step  2  :

 2.1    Rewrite   (y-x)    as  (-1) •  (x-y) 

Evaluate an expression :

 2.2    Multiply  (x-y)  by  (x-y) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x-y)  and the exponents are :
          1 , as  (x-y)  is the same number as  (x-y)¹ 
 and   1 , as  (x-y)  is the same number as  (x-y)¹ 
The product is therefore,  (x-y)⁽¹⁺¹⁾ = (x-y)² 

Equation at the end of step  2  :

  ((((x2)•(y-z))+((y2)•(z-x)))+((z2)•(x-y)))-(x-y)2•(y-z)  = 0 
 Step  3  :

Equation at the end of step  3  :

  ((((x2)•(y-z))+((y2)•(z-x)))+z2•(x-y))-(x-y)2•(y-z)  = 0 
 Step  4  :

Equation at the end of step  4  :

  ((((x2)•(y-z))+y2•(z-x))+z2•(x-y))-(x-y)2•(y-z)  = 0 
 Step  5  :

Equation at the end of step  5  :

  ((x2•(y-z)+y2•(z-x))+z2•(x-y))-(x-y)2•(y-z)  = 0 

Step  6  :

 6.1     Evaluate :  (x-y)2   =    x2-2xy+y2 

 6.2    Factor  xy²-2xyz+xz²-y³+2y²z-yz² 

Try to factor this 6-term polynomial into (2-term) • (3-term)

Begin by splitting the 6-term into two 3-term polynomials:

 xy²-2xyz+xz²  and  -y³+2y²z-yz² 

Next simplify each 3-term polynomial by pulling out like terms:

 x • (y²-2yz+z²)  and  -y • (y²-2yz+z²) 

Note that the two simplified polynomials have  y²-2yz+z²  in common

Now adding the two simplified polynomials we get

 (-y+x)  •  (y²-2yz+z²) 

Which is the desired factorization.

Trying to factor a multi variable polynomial :

 6.3    Factoring    y² - 2yz + z² 

Try to factor this multi-variable trinomial using trial and error 

 Found a factorization  :  (y - z)•(y - z)

Detecting a perfect square :

 6.4    y²  -2yz  +z²  is a perfect square 

 It factors into  (y-z)•(y-z)
which is another way of writing  (y-z)²

How to recognize a perfect square trinomial:  

 • It has three terms  

 • Two of its terms are perfect squares themselves  

 • The remaining term is twice the product of the square roots of the other two terms

Equation at the end of step  6  :

  (x - y) • (y - z)2  = 0 

Step  7  :

Theory - Roots of a product :

 7.1    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.

Equation of a Straight Line

 7.2     Solve   x-y  = 0

Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).

"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.

In this formula :

y tells us how far up the line goes
x tells us how far along
m is the Slope or Gradient i.e. how steep the line is
b is the Y-intercept i.e. where the line crosses the Y axis

The X and Y intercepts and the Slope are called the line properties. We shall now graph the line  x-y  = 0 and calculate its properties

Graph of a Straight Line :

Calculate the Y-Intercept :

Notice that when x = 0 the value of y is 0/-1 so this line "cuts" the y axis at y=-0.00000

  y-intercept = 0/-1  = -0.00000 

Calculate the X-Intercept :

When y = 0 the value of x is 0/1 Our line therefore "cuts" the x axis at x= 0.00000

  x-intercept = 0/1  =  0.00000 

Calculate the Slope :

Slope is defined as the change in y divided by the change in x. We note that for x=0, the value of y is 0.000 and for x=2.000, the value of y is 2.000. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of 2.000 - 0.000 = 2.000 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

     Slope     = 1

Solving a Single Variable Equation :

 7.3     Solve   y-z  = 0

In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.

We shall not handle this type of equations at this time.

Geometric figure: Straight Line

1.   Slope = 1
2.   x-intercept = 0/1 = 0.00000
3.   y-intercept = 0/-1 = -0.00000