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.