Section 2.1
Common Factors
Factoring polynomial expressions is not exactly the same as factoring numbers, but the ideas and techniques are very similar. When factoring numbers or factoring polynomials (expressions), you are finding numbers or polynomials (expression) that divide out evenly from the original numbers or polynomials. But in the case of polynomials, you are dividing numbers and variables out of expressions.Factoring is backwards from distributing. That is, instead of multiplying something through a parentheses, you will be looking for what you can take back out and put in front of a parentheses, such as in the following examples.
Examples:
a. $6x^3+4x^2=2x^2(3x+2)$ |
The $2x^2$ is a common factor in both $6x^3$ and $4x^2$. |
b. $3x(a+b)-5(a+b)=(a+b)(3x-5)$ | The
$(a+b)$ is a common factor in both $3x(a+b)$ and $-5(a+b)$. |
c. $3x(a+b)-(a+b)=(a+b)(3x-1)$ | Again,
$(a+b)$ is the common factor in both experessions. |
The hardest part is to see what can be factored out of every term in the expression.
Exercises:
Factor out the common factor in each of the following.
\begin{array}{l l}
1. \;\;2{{x}^{2}}-4x\;\;& 2. \;\;4{{x}^{3}}+12x\;\;\\
3. \;\;2{{x}^{4}}-6x\;\;& 4. \;\;3{{x}^{10}}-27{{x}^{11}}\;\;\\
5. \;\;{{x}^{6}}+{{x}^{4}}\;\;& 6. \;\;{{x}^{2}}-2x\;\;\\
7. \;\;5{{x}^{4}}-3{{x}^{3}}+2{{x}^{2}}\;\;& 8. \;\;2{{x}^{2}}-8x\;\;\\
9. \;\;6{{x}^{3}}+4{{x}^{2}}\;\;& 10. \;\;36{{x}^{2}}-24x+18{{x}^{3}}\;\;\\
11. \;\;6x{{y}^{2}}-8{{x}^{2}}y\;\;& 12. \;\;3x(a+b)-5(a+b)\;\;\\
13. \;\;3{{x}^{2}}(a+b)-6{{x}^{3}}(a+b)\;\;& 14. \;\;{{x}^{2}}(a+b)-6x(a+b)+7(a+b)\;\;\\
15. \;\;3x(a+b)-(a+b)\;\;& 16. \;\;6{{x}^{3}}{{y}^{2}}z-12{{x}^{2}}y{{z}^{3}}+18x{{y}^{3}}{{z}^{2}}\;\;\\
17. \;\;6{{x}^{2}}(P-Q)-12x(P-Q)\;\;& 18. \;\;7x{{(P+Q)}^{2}}+3(P+Q)\;\;\\
19. \;\;7x{{(P+Q)}^{2}}+(P+Q)\;\;& 20. \;\;(x+y)(p-q)-(x+y)(p+q)\;\;\\
21. \;\;(x+y)(p-q)+(x-y)(p-q)\;\;& 22. \;\;(x+y)(p-q)+(x+y)(p-q)\;\;\\
23. \;\;(x+y)(p+q)-(x+y)\;\;& 24. \;\;(x+y)(p+q)+(p+q)\;\;
\end{array}