# 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 1Factor completely the polynomial4 x^{ 2} + 4 x + 3 x + 3Note 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 2Factor the polynomial2 x^{ 2} - 4 x + 3 x y - 6 yThere 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 3Factor the polynomialx y - x - 2 y + 2Note 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 4Factor completely the polynomial 3 x^{ 2} + 4 x + 1Note 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 + 13 x^{ 2} + 3 x + x + 1We 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 completelyDetailed Solutions and explanations to these questions. a) 2 x^{ 2}- 4 x + x y - 2 y b) x^{ 2} + 3 x - 2 x - 6c) 15 x^{ 2} - 3 x + 10 x - 2 d) 4 x^{ 2} + x - 3e) x^{ 2} y + 3 x + x^{ 2} y^{ 2} + 3 x yf) 3 x^{ 2} + 3 x y - x + 2 y - 2Detailed Solutions and explanations to these questions. |

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