please clear my algebra concept

Question

math doest allow coefficient & their variables cross out when they are subtracted and added.

for ex( 2x^2+6x)/(2x^2+7x+3)
2x^2&2x^2 we can not cross out because they are added

next ex
(21x^3y-6x^2y^4+3xy)/3xy
in this ex how can we divide 21x^3y by 3xy and next step will come 7x^2
21x^3y this has subtracted. so how?
may be only in subtraction coefficient & their variables we can cross out is legal

please can you explain me concept 

Answer 1

1.  (2x²+6x)/(2x²+7x+3)

 

We can factor the numerator and denominator to see

if they have factors in common that can be cancelled.

 

Numerator:  2x²+6x

Both are divisible by 2x so factor that out

2x²+6x = 2x(x+3)

 

Denominator: 2x²+7x+3

Factor by grouping.  Multiply the coefficient of the x² term, 2,

by the constant, 3.  2(3)=6

Look for factors of 6 that add to the coefficient of the x term, 7

6(1)=6 and 6+1=7

Replace 7x with 6x+x

2x²+7x+3

= 2x²+6x+x+3

Factor 2x out of the first two terms

=2x(x+3)+x+3

 

=2x(x+3) + 1(x+3)

= (2x+1)(x+3)

 

Now your fraction is:

 

[2x(x+3)]/[(2x+1)(x+3)]

 

Both numerator and denominator have the factor (x+3)

in common so that can now be cancelled.

 

= 2x/(2x+1)

 

2.  (21x³y-6x²y⁴+3xy)/3xy

 

We can see that 3xy/3xy = 1 so separate this into two fractions

Just as   (9+4)/9 = 9/9 + 4/9 = 1 + 4/9 = 1 4/9

 

(21x³y-6x²y⁴ + 3xy)/3xy = (21x³y-6x²y⁴)/3xy + 3xy/3xy

 

= (21x³y-6x²y⁴)/3xy + 1

 

We can factor 3x²y out of each term in the numerator

 

= [3x²y(7x-2y³)]/3xy + 1

 

We can now cancel common factors from numerator and denominator.

The 3, the x and the y cancel...  so cancel 3xy out of numerator & denominator

 

= x(7x-2y³) + 1

 

********************************************

 

Note:  You could have factored just 3xy out of the numerator

 

[3xy(7x²-2xy³)]/3xy + 1 = 7x²-2xy³ + 1

 

This is the same answer as the previous just without the

x factored out of the first two terms.

 

 

Answer 2

for ex( 2x^2+6x)/(2x^2+7x+3) = 2x(x+3) / [2x+1)(x+3)] = 2x / (2x+1) ... simplified by factoring; identical factors of numerator & denominator cancel.

 

(21x^3y-6x^2y^4+3xy)/3xy is division of a polynomial by a monomial. The denominator is divided into ech term, separately (repeatedly, you might say), resulting in  = 7x² - 2x + 1.

 

Check this work for yourself (just in case I mistyped).

 

Cancellations can only occur when numerator & denominator are fully factored, when polynomials divide.  And monomial denominators are common denominators.