Factoring by grouping
Just like it says, factoring by grouping means that you will group terms with common factors before factoring.
This is done by grouping a pair of terms. Then, factor each pair of two terms.
I will illustrate this with a simple example
Factor x
^{2} + 5x + 6
The expression x
^{2} + 5x + 6
has three terms right now, so we need to write it with 4 terms
before we can group terms.
Notice that 5x = 3x + 2x, so x
^{2} + 3x + 2x + 6
Group x
^{2} with 3x and 2x with 6 and then factor each group.We get:
(x
^{2} + 3x) + (2x + 6) = x*(x + 3) + 2*(x + 3) = (x + 3) * (x + 2)
In this example, if you group x
^{2} with 2x and 3x with 6, you will get the same answer. Try doing that.
Notice that there is more than one way we can expand 5x, so different groupings are possible.
5x is also equal to 4x + x, 6x -x, 7x-2x, 8x-3x, and so forth...
However, only one grouping will work
This brings light to the fact that this way of factoring by grouping can be very tedious sometimes.
Although it is always good to know, it is not always a straightforward
method to factor trinomials
Example #2:
x
^{2}+ -4x + -12
At first, you may be tempted to say that -4x can be equal to: -2x + -2x, or -3x + -x, so one of them will work
Wrong! The right combination is -6x + 2x
So, x
^{2}+ -4x + -12 = x
^{2} + -6x + 2x + -12
Group x
^{2} with -6x and 2x with -12
(x
^{2} + -6x) + (2x + -12) = x *(x − 6) + 2 * (x − 6)= (x − 6)*(x + 2)
Example #3:
3y
^{2} + 14y + 8
3y
^{2} + 14y + 8 = 3y
^{2} + 12y + 2y + 8 = (3y
^{2} + 12y) + (2y + 8) = 3y(y + 4) + 2(y + 4)
So, 3y
^{2} + 14y + 8 =(y + 4)(3y + 2)
Example #4:
11x
^{2} + -41x + -12
This problem is very complicated because you have too many choices for things you can add to get -41x
Some possibilities are:
.....
.....
-46x + 5x
-45x + 4x
-44x + 3x
-40x + -1x
-39x + -2x
-38x + -3x
-36x + -4x
.....
.....
It turns out that the right combination is - 44x + 3x
There is good news though since there is a technique to use to find the right combination a little bit faster when factoring by grouping.
Do 11 * -12 = -132
Then, find factors of -132 that will add up to -41
The factors are -44 and 3
11x
^{2} + -41x + -12 = 11x
^{2} + -44x + 3x + -12
11x
^{2} + -44x + 3x + -12 = 11x(x − 4) + 3(x − 4) = (x − 4)(11x + 3)
Example #5:
6x
^{2}- 26x + 28
6 * 28 = 168
-14 + -12 = -26 and -14 * -12 = 168, so the right combnation is
6x
^{2}- 26x + 28 = 6x
^{2} + -14x + -12x + 28
6x
^{2} + -14x + -12x + 28 = (6x
^{2} + -14x) + (-12x + 28)= 2x(3x + -7) + -4(3x + -7)
6x
^{2}- 26x + 28 = (3x + -7) * (2x + -4)
Conclusion:
Use factoring by grouping only if you have no other choices!