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.

More References and links

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