Question 152225: factor: x^2-5x+6
thanks
Answer by nabla(475)   (Show Source):
You can put this solution on YOUR website! x^2-5x+6=(x-3)(x-2)
Explanation:
A 2nd degree polynomial (ax^2+bx+c) will factor, if factorisable, to form (1) (dx+e)(fx+g). Now, if we apply foil to (1), we get eg+(ef+dg)x+df x^2
Now,
eg=c
ef+dg=b
df=a
In words, we want two numbers e,g such that e*g equal the constant term of the original polynomial.
Moreover, we want numbers d,f such that d*f equals the coefficient of the x^2 term in the original polynomial.
Finally, we want two numbers f,d such that ef+dg equals the coefficient of the x term in the original polynomial.
As you can see, e and g give us the most useful information. We find e and g by looking at the factors of c.
For your polynomial, c is 6. The factors of 6 are 1,2,3,6.
Now, for your polynomial, a is 1. 1 is the only factor of 1.
So we need numbers out of 1,2,3,6 that add to equal -5. We can see right away that -2+-3=-5. So these must be our f and g. So it factors as (x-3)(x-2)
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