Yes ,we always apply distribution property reason just see here
Let P(x) is polynomial and given as
and Q(x) is monomial given as
`Q(x)xx P(x)=3x^2 xx(2x^2+3x+1)`
`=3x^2 xx2x^2+3x^2 xx 3x+3x^2`
`` Distribution property is always important.
Take utmost care while removing brackets a,d minus sign.
Yes the lonley case you havent to, is when multiply two monomials:
`2xy xx 3x^2y^3= 6x^3 y^4`
In other case distribution rule is to be used:
`3x xx (5x+4y)= 3x xx 5x+ 3x xx 4y= 15x^2+ 12xy`
And more if you have to multiply two polinomials:
`(4x+3y)(7x+5y)= 4x xx 7x+ 4x xx 5y+ 3y xx 7x + 3y xx 5y=`
`=28x^2 +20xy+ 21xy +15y^2` `= 28x^2+41xy+15y^2`
Sometimes it is useful to find to write the polynomial product in a convenient wayinstead to develope by distribution rule:`(x-1)(3x-1)+1=3x `
Supose you have to find solution of this:
Then you may develope it:
You have now a quadratic equation to solve that gives roots as:
`x_1=2` and `x_2=1/3`
Now go back at initial equation:
Inastead to develop product you can subtrac 1 both sides:
Look! the factor 3x -1 is both sides so if you bring it into left sides of equation:
`(x-1)(3x-1) - 1xx (3x-1)=0`
Using inverse distribution rule you get:
Now a product is zero if and oly if one or both terms of the product itself are zero:
So the possible solution are : `x_1=2` and `x_2=1/3`
The same we have find by solution of quadratic equation.
The difference is that in the second method we have solved equation, without using second degreee method, more difficult.
Of course isn't always allowed to do this, but there isn't a unic system to understand to act, but experience, and the more excercise to have a good look about.