He Common factor by grouping Is a way of factorizing, through which the terms of a polynomial are"grouped"to create a more simplified form of the polynomial.
Factorization is a mathematical method that is used to write the polynomials as if they were the product of two or more polynomials. This process is the inverse of the multiplication of polynomials.
An example of factorization by clustering is 2 × 2 + 8x + 3x + 12 is equal to the factorized form (2x + 3) (x + 4).
In clustering factorization, we look for common factors between the terms of a polynomial and, later, distributive property is applied to simplify the polynomial; This is why it is sometimes referred to as a common factor by grouping.
Steps to factorize by clustering
Step # 1
Be sure that the polynomial has four terms; In case it is a trinomial (with three terms), it must be transformed into a polynomial of four terms.
Step # 2
Determine if all four terms have a common factor. If so, we must extract the common factor and rewrite the polynomial.
For example: 5 × 2 + 10 x + 25 × 5
Common factor: 5
5 (x2 + 2x + 5x + 1)
Step # 3
In case the common factor of the first two terms differs from the common factor of the last two terms, the terms must be grouped together with common factors and rewrite the polynomial.
For example: 5 × 2 + 10 × + 2 × + 4
Common factor in 5 × 2 + 10 x: 5x
Common factor in 2x + 4: 2
5x (x + 2) + 2 (x + 2)
Step # 4
If the resulting factors are identical, the polynomial is rewritten including the common factor only once.
For example: 5 × 2 + 10 × + 2 × + 4
5x (x + 2) + 2 (x + 2)
(5x + 2) (x + 2)
Examples of grouping factorization
Example # 1: 6 × 2 + 3x + 20x + 10
This is a polynomial that has four terms, among which there is no common factor. However, terms one and two have 3x as a common factor; While terms three and four have 10 as a common factor.
In extracting the common factors from each pair of terms, we can rewrite the polynomial as follows:
3x (2x + 1) + 10 (2x + 1)
Now, we can see that these two terms have a common factor: (2x + 1); This means that you can extract this factor and rewrite the polynomial again:
(3x + 10) (2x + 1)
Example # 2: x2 + 3x + 2x + 6
In this example, as in the previous one, the four terms do not have a common factor. However, the first two terms have x as common factor, while in the latter two the common factor is 2.
In this sense, you can rewrite the polynomial as follows:
X (x + 3) + 2 (x + 3)
Now, we extract the common factor (x + 3), the result will be as follows:
(X + 2) (x + 3)
Example # 3: 2y3 + y2 + 8y2 + 4y
In this case, the common factor between the first two terms is y2, while the common factor in the last two is 4y.
The rewritten polynomial would be as follows:
Y2 (2y + 1) + 4y (2y + 1)
Now, we extract the factor (2y + 1) and the result is as follows:
(Y2 + 4y) (2y + 1)
Example No. 4: 2 × 2 + 17 × 30
When the polynomial does not present four terms, but is a trinomial (which has three terms), it is possible to factorize by grouping.
However, it is necessary to divide the term of the medium so that it can have four elements.
In the 2 × 2 + 17x + 30 trinomial, the term 17x must be divided into two.
In trinomials that follow the form ax2 + bx + c, the rule is to find two numbers whose product is a x c and whose sum is equal to b.
This means that in this example you need a number whose product is 2 x 30 = 60 and that sum 17. The answer for this is exercise is 5 and 12.
Then we rewrite the trinomial as a polynomial:
2x2 + 12x + 5x + 30
The first two terms have x as common factor, while the common factor in the last two is 6. The resulting polynomial would be:
X (2x + 5) + 6 (2x + 5)
Finally, draw the common factor in these two terms; The result is as follows:
(X + 6) (2x + 5)
Example # 5: 4 × 2 + 13 × + 9
In this example, we also have to divide the middle term to form a four-term polynomial.
In this case, two numbers are needed whose product is 4 x 9 = 36 and whose sum is equal to 13. In this sense, the required numbers are 4 and 9.
Now, the trinomial is rewritten as a polynomial:
4x2 + 4x + 9x + 9
In the first two terms, the common factor is 4x, while in the latter, the common factor is 9.
4x (x + 1) + 9 (x + 1)
Once we extract the common factor (x + 1), the result will be as follows:
(4x + 9) (x +1)
Example # 6: 3 × 3 - 6 × 15 × 30
In the proposed polynomial, all terms have a common factor: 3. Then the polynomial is rewritten as follows:
3 (x3-2x + 5x10)
Now we proceed to group the terms within the parentheses and determine the common factor between them. In the first two, the common factor is x, while in the last two is 5:
3 (x2 (x - 2) + 5 (x - 2))
Finally, the common factor (x - 2) is extracted; The result is as follows:
3 (x2 + 5) (x - 2)
References
- Factoring by grouping. Retrieved on May 25, 2017, from khanacademy.org.
- Factoring: Grouping. Retrieved on May 25, 2017, from mesacc.edu.
- Factoring by grouping examples. Retrieved on May 25, 2017, from shmoop.com.
- Factoring by grouping. Retrieved on May 25, 2017, from basic-mathematics.com.
- Factoring by grouping. Retrieved on May 25, 2017, from https://www.shmoop.com
- Introduction to grouping. Retrieved on May 25, 2017, from khanacademy.com.
- Practice problems. Retrieved on May 25, 2017, from mesacc.edu.