Factoring a polynomial by grouping is explained using several example.

__Example 1:__ Factor completely the polynomial *4 x*^{ 2} + 4 x + 3 x + 3

Note that there is no common factor to all 4 terms in the given polynomial.

But if you group the first two term, you can factor *4 x * out as follows:

*4 x*^{ 2} + 4 x = 4 x (x + 1)

Now if you group the last two term, you can factor *3 * out as follows:

*3 x + 3 = 3 ( x + 1)*

Note that in grouping the terms, we have managed to have *( x + 1)* as a common factor. Rewrite the given polynomial with the grouped terms in factored form.

*4 x*^{ 2} + 4 x + 3 x + 3 = 4 x (x + 1) + 3 ( x + 1)

Note that *( x + 1)* is a common factor which can be factored out as follows:

*4 x*^{ 2} + 4 x + 3 x + 3 = 4 x (x + 1) + 3 ( x + 1) = (x + 1)(4x + 3)

__Example 2:__ Factor the polynomial *2 x*^{ 2} - 4 x + 3 x y - 6 y

There is no common factor to all 4 terms in the given polynomial.

Group the first two terms and factor *2 x* out :

*2 x *^{ 2} - 4 x = 2 x ( x - 2)

Group the last two terms and factor *3 y* out :

*3 x y - 6 y = 3 y( x - 2)*

Note that *( x - 2)* is a common factor. Rewrite the given polynomial with the grouped terms in factored form.

*2 x *^{ 2} + 2 x + 3 x y + 3 y = 2 x (x - 2) + 3 y(x - 2)

Factor out the common factor *(x - 2)* and rewrite the given polynomial in factored form.

*2 x *^{ 2} + 2 x + 3 x y + 3 y = 2 x ( x - 2) + 3 y( x - 2) = (x - 2)(2 x + 3 y)

__Example 3:__ Factor the polynomial *x y - x - 2 y + 2*

Note that there is no common factor to all 4 terms in the given polynomial.

Group the first two terms and factor *x* out:

*x y - x = x ( y - 1)*

Group the last two terms and factor *2* out:

*- 2 y + 2 = 2( - y + 1) = - 2(y - 1)*

Note that *( y - 1)* is a common factor. Rewrite the given polynomial with the grouped terms in factored form.

*x y - x - 2 y + 2 = x (y - 1) - 2 (y - 1) *

Factor out the common factor *(y - 1)* to factor completely

*x y - x - 2 y + 2 = (y - 1) - 2 (y - 1) = (y - 1)(x - 2)*

__Example 4:__ Factor completely the polynomial 3* x*^{ 2} + 4 x + 1

Note tha we have 3 terms only and that there is no common factor to all 3 terms in the given polynomial. One way is to rewrite the polynomial with 4 terms that may be factored by grouping.

We use the identity *4 x = 3 x + x* to rewrite the polynomial as follows:

*3 x *^{ 2} + 4 x + 1 = *3 x*^{ 2} + 3 x + x + 1

We group the first two terms and factor *3 x* out as follows:

*3 x*^{ 2} + 3 x = 3 x (x + 1)

*( x + 1)* is a common factor. Rewrite the given polynomial with the grouped terms in factored form.

*3 x*^{ 2} + 4 x + 1 = 3 x^{ 2} + 3 x + x + 1 = 3 x (x + 1) + 1( x + 1)

Note that *( x + 1)* is a common factor which can be factored out as follows:

*3 x*^{ 2} + 4 x + 1 = 3 x (x + 1) + 1( x + 1) = (x + 1)(3 x + 1)

Use grouping to factor the following polynomials completely.

Detailed Solutions and explanations to these questions.

a) * 2 x*^{ 2}- 4 x + x y - 2 y

b) *x*^{ 2} + 3 x - 2 x - 6

c) * 15 x*^{ 2} - 3 x + 10 x - 2

d) * 4 x*^{ 2} + x - 3

e) *x*^{ 2} y + 3 x + x^{ 2} y^{ 2} + 3 x y

f) *3 x*^{ 2} + 3 x y - x + 2 y - 2

Detailed Solutions and explanations to these questions.