Find all roots of x^3 + 3x^2 - 10x + 6

[stat643]

find all roots of x^3 + 3x^2 - 10x + 6

the solution:
identify the easy root of x=1,
find the remaining roots from (x-1)(x^2+4x) using quadratic formula.

The only thing i don't understand here is how to factorize to (x-1)(x^2+4x)... namely the (x^2+4x) part.

 

[rocomath]

factor out a common term in x^2+4x

and you're pretty much done

 

[stat643]

but how did i get to x^2+4x in the first place?... the original equation was x^3 + 3x^2 - 10x + 6.. i merely copied the solution... so i find the easy root of 1.. then what?

 

[rocomath]

use synthetic division

 

[matt grime]

No, don't use synthetic division (just yet). Pause for a moment and think: is it plausible that x^2+4x is a factor? It isn't. Copying out the answer is never a good idea.

 

[stat643]

i just looked up synthetic devision on wikipedia and tried it but it didnt work

 

[stat643]

No, don't use synthetic division (just yet). Pause for a moment and think: is it plausible that x^2+4x is a factor? It isn't. Copying out the answer is never a good idea.

should i take out the common term x first?

 

[arildno]

i just looked up synthetic devision on wikipedia and tried it but it didnt work

Then you should practice synthetic division once more!

Further, ponder over matt grime's words:

WHY should you be suspicious of that particular factorization?

Hint:
How could you ascertain whether the factorization is correct or false?

 

[stat643]

oh sorry i copied it wrong, it should be (x-1)(x^2+4x-6).. now expanding that get: x^3 + 4x^2 -6x -x^2 -4x + 6 = x^3 + 3x^2 - 10x + 6.. so yeh its right now.. though i still can't get the synthetic devision right (its new to me)

i tried to learn it now from http://en.wikipedia.org/wiki/Synthetic_division

though i keep getting 1,2,-12,18

can someone help show how i would use synthetic devision for the original polynomial ?

 

[arildno]

Okay, we wish to find a second-order polynomial so that:
$$(x-1)(ax^{2}+bx+c)=x^{3}+3x^{2}-10x+6$$ holds for all x.
I.e, we must determine a,b and c!

Multiplying out the left-hand side, and organizing in powers of x, the lefthandside can be rewritten as:
$$ax^{3}+(b-a)x^{2}+(c-b)x-c= x^{3}+3x^{2}-10x+6$$

NOw, the coefficients of each power must be equal on the right and left sides, yielding the system of equations:
a=1
b-a=3
c-b=-10
-c=6

This yields:
a=1
b=4
c=-6

 

[stat643]

thanks arildno, very helpful