How do you factor by grouping?

Question

Example:

x^3 + 3x^2 + 6x + 18

5a^3 - 10a^2 +25a- 50

t^3 + 6t^2 - 2t - 12

x^3 - x^2 -6x + 6

Answer 1

The way to factor by grouping is: look for commonality between the two right-hand terms, and the two left-hand terms. Then you factor it out, and use the distributive property to combine the factors.

For example, your first problem is:

x^3 + 3x^2 + 6x + 18.

In the left-hand pair of terms, the ratio of the coefficients is 1 to 3 (x^3 to x^2). Same with the right-hand pair (6 to 18, which reduces to 1 to 3).

First, separate out the common factor within the pair, leaving only x+3 in each pair:

x^3 + 3x^2 + 6x + 18
x^2 (x + 3) + 6 (x + 3)

Then combine using the distributive property:

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

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In the second problem, the difference is a factor of negative-two between the coefficients of a^3 and a^2 (5 vs -10) and also between a^1 and a^0 (25 vs -50). So you are going to factor out the common factors within each pair, leaving a-2:

5a^3 - 10a^2 + 25a - 50
5a^2 (a - 2) + 25 (a - 2)
(5a^2 + 25)(a - 2)

And there's a factor of 5 which can also be moved out on its own:

5(a^2 + 5)(a - 2)

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In the third problem the ratio is 1 to 6, so you're going to factor each pair down to t+6:

t^3 + 6t^2 - 2t - 12
t^2 (t + 6) + -2(t + 6)
(t^2 - 2)(t + 6)

If you wanted to, you could factor that further as:

(t + sqrt(2))(t - sqrt(2))(t + 6)

... but that's not the point of the exercise.

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In the final problem, the ratio is 1 to -1, so you'll factor down to x-1:

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

Again, that can be factored further:

(x + sqrt(6))(x - sqrt(6))(x - 1)

Answer 2

you group them together so the insides of the parenthesis still are the same expression when you factor out certain variables like for the first one it be x^3 + 6x + 3x^2 + 18 = x(x^2 + 6) + 3(x^2 + 6) = (x^2 + 6)(x + 3)

Answer 3

x^3 + 3x^2 + 6x + 18 =
(x^3 + 3x^2) + (6x + 18) group two and two
x^2(x + 3) + 6(x + 3) find common factor within each parenthesis
(x + 3)(x^2 + 6) factor common factor, i.e. (x + 3)

Answer 4

x^3 + 3x^2 + 6x + 18
(x^3+3x^2)+(6x+18)
take out the greatest common factor
x^2(x+3)+6(x+3)
factor out
(x^2+6)+(x+3)
to check answer, foil and you should get the original problem.

Answer 5

A