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Factoring multivariable polynomials

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a^2+5ab+6b^2=0

This solution deals with factoring multivariable polynomials.

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Step by Step Solution

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Step  1  :

Equation at the end of step  1  :

  ((a2) +  5ab) +  (2•3b2)  = 0 

Step  2  :

Trying to factor a multi variable polynomial :

 2.1    Factoring    a2 + 5ab + 6b2 

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

 
Found a factorization  :  (a + 3b)•(a + 2b)

Equation at the end of step  2  :

  (a + 3b) • (a + 2b)  = 0 

Step  3  :

Theory - Roots of a product :

 3.1    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

 3.2     Solve   a+3b  = 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  a+3b  = 0 and calculate its properties

Graph of a Straight Line :

  
 

Calculate the Y-Intercept :

Notice that when a = 0 the value of b is 0/3 so this line "cuts" the b axis at b= 0.00000

  b-intercept = 0/3  =  0.00000 

Calculate the X-Intercept :

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

  a-intercept = 0/1  =  0.00000 

Calculate the Slope :

Slope is defined as the change in b divided by the change in a. We note that for a=0, the value of b is -0.000 and for a=2.000, the value of b is -0.667. So, for a change of 2.000 in a (The change in a is sometimes referred to as "RUN") we get a change of -0.667 - -0.000 = -0.667 in b. (The change in b is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

    Slope     = -0.667/2.000 = -0.333 

Equation of a Straight Line

 3.3     Solve   a+2b  = 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  a+2b  = 0 and calculate its properties

Graph of a Straight Line :

  
 

Calculate the Y-Intercept :

Notice that when a = 0 the value of b is 0/2 so this line "cuts" the b axis at b= 0.00000

  b-intercept = 0/2  =  0.00000 

Calculate the X-Intercept :

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

  a-intercept = 0/1  =  0.00000 

Calculate the Slope :

Slope is defined as the change in b divided by the change in a. We note that for a=0, the value of b is -0.000 and for a=2.000, the value of b is -1.000. So, for a change of 2.000 in a (The change in a is sometimes referred to as "RUN") we get a change of -1.000 - -0.000 = -1.000 in b. (The change in b is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

    Slope     = -1.000/2.000 = -0.500 

Geometric figure: Two Straight Lines

  1.   Slope = -1.000/2.000 = -0.500
  2.   a-intercept = 0/1 = 0.00000
  3.   b-intercept = 0/2 = 0.00000
  4.   Slope = -0.667/2.000 = -0.333
  5.   a-intercept = 0/1 = 0.00000
  6.   b-intercept = 0/3 = 0.00000

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