Answer:
1.) [tex](x-5)(x-1)[/tex]
2.) [tex](x-2)(x+11)[/tex]
3.) Prime
Solution Steps:
______________________________
1.) [tex]\bold{\sf{x^2-6x+5}}[/tex]:
- Factor the expression by grouping. First, the expression needs to be rewritten as [tex]x^2+ax+bx+5[/tex]. To find a and b, set up a system to be solved:
- [tex]a+b=-6[/tex]
- [tex]ab=1*5=5[/tex]
- Since ab is positive, a and b have the same sign. Since [tex]a+b[/tex] is negative, a and b are both negative. The only such pair is the system solution:
- [tex]a=-5[/tex]
- [tex]b=-1[/tex]
- Rewrite [tex]x^2-6x+5[/tex]:
- [tex](x^2-5x)+(-x+5)[/tex]
- Factor out x in the first and −1 in the second group:
- [tex]x(x-5)-(x-5)[/tex]
- Factor out common term [tex]x-5[/tex] by using distributive property:
- [tex](x-5)(x-1)[/tex]
2.) [tex]\bold{\sf{x^2+9x-22}}[/tex]:
- Factor the expression by grouping. First, the expression needs to be rewritten as [tex]x^2+ax+bx-22[/tex]. To find a and b, set up a system to be solved:
- [tex]a+b=9[/tex]
- [tex]ab=1(-22)=-22[/tex]
- Since ab is negative, a and b have the opposite signs. Since [tex]a+b[/tex] is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product −22:
- [tex]-1,22[/tex]
- [tex]-2,11[/tex]
- Calculate the sum for each pair:
- [tex]-1+22=21[/tex]
- [tex]-2+11=9[/tex]
- The solution is the pair that gives sum 9:
- [tex]a=-2[/tex]
- [tex]b=11[/tex]
- Rewrite [tex]x^2+9x-22[/tex]:
- [tex](x^2-2x)+(11x-22)[/tex]
- Factor out x in the first and 11 in the second group:
- [tex]x(x-2)+11(x-2)[/tex]
- Factor out common term [tex]x-2[/tex] by using distributive property:
- [tex](x-2)(x+11)[/tex]
3.) [tex]\bold{\sf{x^2+10x+8}}[/tex]:
- This cannot be factored.
______________________________
