Section 2.4
Trinomials: $x^2\pm bx\pm c$
We consider factoring trinomial expressions of the form $x^2\pm bx\pm c$. The technique is not as straight forward as recognizing a pattern and applying a formula, but there is a sytematic way of attacking these types of problems.
Start by taking both $b$ and $c$ to be positive, but each has an arithmetic sign — plus or minus — associated with it. We start by finding factors of $c$.
- If $c$ has a plus sign, then we need factors that add up to $b$.
- If $c$ has a minus sign, then we need factors whose difference is $b$.
In each case, call the two numbers $p$ and $q$.
We then write down the framework of the factorization: $$(x\;\;\;\;\;\;p)(x\;\;\;\;\;\;q).$$ It is then a matter of filling in appropriate plus and minus signs in each factor.
Examples:
a. To factor $x^2+6x+8$, we note that 8 has two pairs of factors: 1 and 8; 2 and 4.
The 8 has a plus sign, so we need two numbers that add up to 6; they are 2 and 4.
The framework for the factorization is $(x\;\;\;\;\;\;2)(x\;\;\;\;\;\;4)$.
We want a "plus 6" in the middle; both signs should be "plus."
The factored form of $x^2+6x+8$ is $(x+2)(x+4)$.
b. To factor $x^2-8x-20$, we note that 20 has three pairs of factors: 1 and 20; 2 and 10; 4 and 5.
The 20 has a minus sign, so we need two numbers whose difference is 8; they are 2 and 10.
The framework for the factorization is $(x\;\;\;\;\;\;2)(x\;\;\;\;\;\;10)$.
We want a "minus 8" in the middle; the 10 gets a minus and the 2 gets a plus.
The factored form of $x^2-8x-20$ is $(x+2)(x-10)$.
Exercises:
Factor each of the following completely.
\begin{array}{l l l l}
1. \;\;{{x}^{2}}+8x+12\;\;& 2. \;\;{{x}^{2}}+7x+6\;\;& 3. \;\;{{x}^{2}}+7x+12\;\;& 4. \;\;{{x}^{2}}+11x+18\;\;\\
5. \;\;{{x}^{2}}+9x+18\;\;& 6. \;\;{{x}^{2}}+9x+14\;\;& 7. \;\;{{x}^{2}}-3x+2\;\;& 8. \;\;{{x}^{2}}-3x-4\;\;\\
9. \;\;{{x}^{2}}-x-12\;\;& 10. \;\;{{x}^{2}}-10x+21\;\;& 11. \;\;{{x}^{2}}-3x-10\;\;& 12. \;\;{{x}^{2}}-x-20\;\;\\
13. \;\;{{x}^{2}}-10x-24\;\;& 14. \;\;{{x}^{2}}+7x+10\;\;& 15. \;\;{{x}^{2}}-9x+20\;\;& 16. \;\;{{x}^{2}}+8x-20\;\;\\
17. \;\;{{x}^{2}}+3x-10\;\;& 18. \;\;{{x}^{2}}-21x+20\;\;& 19. \;\;{{x}^{2}}+5x-6\;\;& 20. \;\;{{x}^{2}}+5x+6\;\;\\
21. \;\;{{x}^{2}}-5x+6\;\;& 22. \;\;{{x}^{2}}-5x-6\;\;& 23. \;\;{{x}^{2}}+10x-39\;\;& 24. \;\;{{x}^{2}}-5x-36\;\;\\
25. \;\;{{x}^{2}}-12x-28\;\;& 26. \;\;{{x}^{2}}-11x+28\;\;& 27. \;\;{{x}^{2}}+3x-4\;\;& 28. \;\;{{x}^{2}}-8x+15\;\;\\
29. \;\;{{x}^{2}}+5x-24\;\;& 30. \;\;{{x}^{2}}+2x-24\;\;& 31. \;\;{{x}^{2}}-7x-18\;\;& 32. \;\;{{x}^{2}}+17x-60\;\;\\
33. \;\;{{x}^{2}}-19x+60\;\;& 34. \;\;{{x}^{2}}-19x-20\;\;& 35. \;\;{{x}^{2}}-13x+22\;\;& 36. \;\;{{x}^{2}}-5x-84\;\;\\
37. \;\;{{x}^{2}}+14x+33\;\;& 38. \;\;{{x}^{2}}-8x+15\;\;& 39. \;\;{{x}^{2}}+13x+12\;\;& 40. \;\;{{x}^{2}}+15x-100\;\;\\
41. \;\;{{x}^{2}}+5x-50\;\;& 42. \;\;{{x}^{2}}+11x+24\;\;& 43. \;\;{{x}^{2}}-13x+40\;\;& 44. \;\;{{x}^{2}}-13x+42\;\;\\
45. \;\;{{x}^{2}}-3x-88\;\;& 46. \;\;{{x}^{2}}-18x+77\;\;& 47. \;\;{{x}^{2}}+13x+36\;\;& 48. \;\;{{x}^{2}}+2x-48\;\;
\end{array}