Section 2.5
Grouping
We consider now certain four-term expressions that can be factored if we group the terms appropriately.
Examples:
The first way of grouping a four-term expression is into two groups with two terms in each---"2 by 2." The aim of this grouping is to factor each of those two-term expressions and then to discover that there is a common factor that results. Note in examples (a.) and (b.) below that in the second line, the expression is still a sum; it has not yet been written in factored form. However, the two terms in that sum have a common factor.
a. \begin{align}
ax+2a+cx+2c&=(ax+2a)+(cx+3c)&&\textit{Group the first two terms and the last two terms together.}\\
&=a(x+2)+c(x+2)&&\textit{Factor a common factor from each grouping.}\\ &=(x+2)(a+c)&&\textit{Factor a common factor from both expressions.}
\end{align}
b. \begin{align}
2kx-4-akx+2a&=(2kx-4)-(akx-2a)&&\textit{Group the first two terms together.}\\
&&&\textit{Be carefull inserting parenthesis in front of a minus sign.}\\
&=2(kx-2)-a(kx-2)&&\textit{Factor a common factor from each grouping.}\\
&=(2-a)(kx-2)&&\textit{Factor a common factor from both expressions.}
\end{align}
Note also in example (b.) the care that must be taken when the third term is preceded by a minus sign. It is sometimes possible to get around that problem by rearranging the terms. In examples (c.) and (d.) below, the same expression is factored—with the same final result, but going at the problem in two different ways.
c. \begin{align}
2Kx+3x-2aK-3a&=(2Kx+3x)-(2aK+3a)\\
&=x(2K+3)-a(2K+3)\\
&=(x-a)(2K+3)
\end{align}
d. \begin{align}
2Kx+3x-2aK-3a&=2Kx-2aK+3x-3a\\
&=(2Kx-2aK)+(3x-3a)\\
&=2K(x-a)+3(x-a)\\
&=(x-a)(2K+3)
\end{align}
The second way to group a four-term expression is to group "three and one" or "one and three." The purpose here is to get a three-term expression that is a perfect square and a one-term expression that is also a perfect square, with a minus sign between the two groups. That way the four-term expression can be factored as the difference of two squares.
e. \begin{align}
x^2+8x+16-4t^2&=\left(x^2+8x+16\right)-4t^2&&\textit{Group the first three terms together}.\\
&=(x+4)^2-(2t)^2&&\textit{Factor the first three terms as a trinomial square.}\\
&&&\text{The second term is also a perfect square.}\\
&=\left[(x+4)+2t)\right] \left[(x+4)-2t)\right]&&\textit{Factor as the difference of squares.}
\end{align}
f.\begin{align}
4x^2-4xy+y^2-64&=\left(4x^2-4xy+y^2\right)-64\\
&=(2x-y)^2-8^2\\
&=\left[(2x-y)+8\right] \left[(2x-y)-8\right]
\end{align}
Note that for this kind of problem, the original expression has three perfect squares, but two of them go together, with the remaining term as the "middle term" to make up the three-term perfect square. Here is an example of factoring a four-term expression by grouping "one and three." Again, minus signs can cause difficulties.
g. \begin{align}
25-x^2+4xy-4y^2&=25-\left(x^2-4xy+4y^2\right)\\
&=5^2-(x-2y)^2\\
&=\left[5+(x-2y)\right] \left[5-(x-2y)\right]
\end{align}
Example 1: Basic grouping: Factoring Trinomials by Grouping 1
Example 2: Basic grouping 2x2: Factoring Trinomials by Grouping 1
Exercises:
Factor each of the following completely
\begin{array}{l l}
1. \;\;8{{a}^{2}}-2a-1\;\;& 2. \;\;Kx+5Kt+xy+5ty\;\;\\
3. \;\;xy+7y+3x+21\;\;& 4. \;\;{{x}^{2}}+36\;\;\\
5. \;\;{{x}^{2}}+4xy+3Kx+12Ky\;\;& 6. \;\;6{{x}^{2}}+15x+4xy+10y\;\;\\
7. \;\;3{{x}^{2}}-6xy+4x-8y\;\;& 8. \;\;rx+rz\;\;\\
9. \;\;4{{x}^{3}}y-8{{x}^{2}}{{y}^{2}}-4xy+8{{y}^{2}}\;\;& 10. \;\;{{x}^{2}}-{{y}^{2}}-x+y\;\;\\
11. \;\;{{4x}^{2}}+4x+1-t^2\;\;& 12. \;\;{{x}^{2}}-6x+9-25{{K}^{2}}\;\;\\
13. \;\;3{{x}^{2}}-6x+3-27{{y}^{2}}\;\;& 14. \;\;9{{x}^{2}}-{{y}^{2}}-8y-16\;\;\\
15. \;\;5x+15-ax-3a\;\;& 16. \;\;12{{K}^{2}}-12{{x}^{2}}+12xy-3{{y}^{2}}\;\;\\
17. \;\;36-{{x}^{2}}-10xy-25{{y}^{2}}\;\;& 18. \;\;64{{x}^{2}}-{{y}^{2}}-4Ky-4{{K}^{2}}\;\;\\
19. \;\;2{{x}^{2}}-18+24y-8{{y}^{2}}\;\;& 20. \;\;{{x}^{2}}-2xy+{{y}^{2}}-{{K}^{2}}-8K-16\;\;\\
21. \;\;6{{x}^{3}}{{y}^{2}}+15{{x}^{2}}yt-4xy{{t}^{2}}-10{{t}^{3}}\;\;& 22. \;\;3{{x}^{2}}-6xy+3{{y}^{2}}-48{{K}^{2}}\;\;\\
23. \;\;{{x}^{2}}-{{y}^{2}}-4x-4y\;\;& 24. \;\;14{{x}^{2}}-49x-28\;\;\\
25. \;\;{{x}^{2}}-4xy+4{{y}^{2}}-64\;\;& 26. \;\;15x-6xy+10y-4{{y}^{2}}\;\;\\
27. \;\;4{{x}^{2}}-8xy+4{{y}^{2}}-16{{K}^{2}}+16K-4\;\;& 28. \;\;15{{x}^{2}}+10xy+9x+6y\;\;\\
29. \;\;9{{x}^{2}}-9{{y}^{2}}+12y-4\;\;& 30. \;\;5{{x}^{2}}y-5xy-10y\;\;
\end{array}