Factor Polynomials by Grouping - Questions With Detailed Solutions

How to factor a polynomial by grouping? First a group of examples with detailed solutions are presented then Grade 11 questions are presented along with detailed solutions.

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.

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Updated: 20 January 2017 (A Dendane)

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