Factoring by grouping

Just like it says, factoring by grouping means that you will group terms with common factors before factoring.

As you can see, this is done by grouping a pair of terms. Then, factor each pair of two terms. If you did not understand the example above, keep reading as we explain the concept with more examples.

More examples enplaning factoring by grouping

Factor x² + 5x + 6

The expression x² + 5x + 6 has three terms right now, so we need to write it with 4 terms before we can group terms.

5x = 3x + 2x, so x² + 5x + 6 becomes x² + 3x + 2x + 6.

Group x² with 3x and 2x with 6 and then factor each group.

We get (x² + 3x) + (2x + 6) = x*(x + 3) + 2*(x + 3) = (x + 3) * (x + 2)

In this example, if you group x² with 2x and 3x with 6, you will get the same answer. Try doing that.

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:

x² + -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, x² + -4x + -12 = x² + -6x + 2x + -12

Group x² with -6x and 2x with -12

(x² + -6x) + (2x + -12) = x *(x − 6) + 2 * (x − 6) = (x − 6)*(x + 2)

Example #3:

3y² + 14y + 8

3y² + 14y + 8 = 3y² + 12y + 2y + 8 = (3y² + 12y) + (2y + 8) = 3y(y + 4) + 2(y + 4)

So, 3y² + 14y + 8 = (y + 4)(3y + 2)

Example #4:

11x² + -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

11x² + -41x + -12 = 11x² + -44x + 3x + -12

11x² + -44x + 3x + -12 = 11x(x − 4) + 3(x − 4) = (x − 4)(11x + 3)

Example #5:

6x²- 26x + 28

6 * 28 = 168

-14 + -12 = -26 and -14 * -12 = 168, so the right combination is

6x²- 26x + 28 = 6x² + -14x + -12x + 28

6x² + -14x + -12x + 28 = (6x² + -14x) + (-12x + 28)= 2x(3x + -7) + -4(3x + -7)

6x²- 26x + 28 = (3x + -7) * (2x + -4)

Conclusion:

Use factoring by grouping only if you have no other choices!

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