Question

To factor atex-formula">+bx+c, you have to group it as (a$x^{2}$+ux)+(vx+c) so that uv=ac and u+v=b. Both addition and multiplication are commutative, so how do you determine which value is assigned to u and which to v? Or are they interchangeable?

Answer 1

Answer: They are interchangeable

For example, let's expand out (x+3)*(2x+1). Use of the FOIL rule should get to you 2x^2 + 7x + 3.

Now imaging that you handed 2x^2 + 7x + 3 to a friend to be factored. They do not know the factorization yet.

They would see that a = 2, b = 7, c = 3.

The friend needs to find two numbers u,v such that they multiply to a*c = 2*3 = 6 and add to b = 7

Through trial and error (I recommend making a table of values), your friend would get

• u*v = 1*6 = 6
• u+v = 1+6 = 7

But as you mentioned, your friend could easily swap the order of addition and multiplication to say

• u*v = 6*1 = 6
• u+v = 6+1 = 7

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If we go with the first option, then the factor by grouping looks like this

2x^2 + 7x + 3

2x^2 + 1x + 6x + 3 ... break up 7x into 1x+6x

(2x^2+1x) + (6x+3) .... group up the terms into pairs

x(2x+1) + 3(2x+1) .... factor each grouping

(x+3)(2x+1) ..... factor out the overall GCF

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Or we could have the factor by grouping looks like this

2x^2 + 7x + 3

2x^2 + 6x + 1x + 3 ... break up 7x as 6x+1x

(2x^2+6x) + (1x+3)

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

(2x+1)(x+3)

(x+3)(2x+1)

We end up with the same factorization.

Answer 2

Answer:

I'm not sure.  Try asking Google.  

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