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Factor a two-variable polynomial by grouping

Group terms that share the same polynomial factor

Factoring by grouping works when separate groups each contain the same remaining expression. First factor that common expression out, then continue factoring it.

Problem

Factor $x^{3} + 3 x^{2} y + x^{2} + 2 x y^{2} + 3 x y + 2 y^{2}$ .

Answer

(x + 1) (x + y) (x + 2 y)

The middle quadratic is x²+3xy+2y².

Step-by-step solution

1. Rearrange into two matching groups

Write the polynomial as

(x^{3} + 3 x^{2} y + 2 x y^{2}) + (x^{2} + 3 x y + 2 y^{2}) .

The first group is $x (x^{2} + 3 x y + 2 y^{2})$ and the second is $1 (x^{2} + 3 x y + 2 y^{2})$ .

2. Take out the common quadratic

The expression becomes $(x + 1) (x^{2} + 3 x y + 2 y^{2})$ .

3. Factor the quadratic

The two terms $y$ and $2 y$ add to $3 y$ and multiply to $2 y^{2}$ . Therefore $x^{2} + 3 x y + 2 y^{2} = (x + y) (x + 2 y)$ , giving the result $(x + 1) (x + y) (x + 2 y)$ .

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