When we have a polynomial of the form
ax^2+bx+c
We can factor this quadratic with splitting up the b term into two terms. This allows us to factor the left side of the expression and the right side individually, and look for a common factor between them. This is factoring by grouping.
Let's take our polynomial
ax^2+bx+c again. To factor by grouping, we can rewrite this expression as
color(blue)(a)x^2+bx+color(red)(c)=color(blue)ax^2+(color(blue)a+color(red)c)x+color(red)c
Notice that (a+c)x is the same as our b term. We can distribute the x to both terms to get
color(blue)(ax^2+ax)+color(red)(cx+c)
This is the essence of factoring by grouping. We can look at our polynomial as two groups of two terms.
From the blue terms, we can factor out an ax, and from the red terms, a c. This leaves us with
color(blue)(ax)color(purple)((x+1))+color(red)c color(purple)((x+1))
Now, both terms have an x+1 in common, so we can factor that out to get
(color(blue)(ax)+color(red)(c))color(purple)((x+1))
We will not always have an x+1 term. For instance, take the following polynomial:
3x^2+11x+6
Let's rewrite this as
color(turquoise)(3x^2+9x)+color(orange)(2x+6)
We can factor a 3x out of the blue terms, and a 2 out of the orange terms. We get
color(turquoise)(3x)(x+3)+color(orange)2(x+3)
We can factor an x+3 out to get
(3x+2)(x+3)
The key point is that we can rewrite our b term as the sum of two terms so we can factor twice. Next, we look for a common factor between our newly factored expression.
Factoring by grouping will not always work- at this point it may be a good idea to resort to the Quadratic Formula.
Hope this helps!
