## sabika13 3 years ago Prove that (x-a) is a factor of x^3 - (a + b + c)x^2 + (ab +bc + ca)x-abc. How do I prove this???

1. satellite73

show that if you replace \(x\) by \(a\) you get 0

2. satellite73

then you know if \(a\) is a root of a polynomial \(p(x)\) then you can factor as \(p(x)=(x-a)(q(x))\)

3. sabika13

yeah i tried that, but i couldnt end up with a zero..

4. satellite73

if you do not get 0, then you cannot factor

5. sabika13

i probably made a mistake.. because it has to equal to zero

6. satellite73

\[a^3 - (a + b + c)a^2 + (ab +bc + ca)a-abc\] is a startr

7. satellite73

yeah it works, try again

8. sabika13

a^3 - (a + b +c)a^2 + (ab+ bc +ca)a -abc = a^3 - a^3 + a^3b + a^3 c + a^2b + bc+ca^ - abc

9. sabika13

ohhh i see it, i messed up

10. satellite73

hmmm forgot the distributive property for the second term

11. satellite73

and some other mistakes too, but you can clean it up i am sure

12. sabika13

okay i tried and i ended up like this: 2a^2 + 2a^2c

13. satellite73

lets go slow

14. sabika13

ohh its negative.. i got it

15. satellite73

first step \[a^3 - (a + b + c)a^2 + (ab +bc + ca)a-abc\] second step ok

16. sabika13

yeah i didnt see the negative sign infront of the first bracket, thank you so much!!

17. satellite73

yw (you did all the work)