Question 347463
To simplify (or reduce) a fraction, you look for common factors to cancel. This has been true since the "good old days" of fractions like 2/4. The only thing different with fractions like {{{(x^2-10x+9)/(x^2+2x-3)}}} is that finding the factors is harder.<br>
Factoring your numerator is a matter of finding the factors of 9 which add up to -10. In order for the product of the factors to be positive 9, they must be both positive <i>or both negative</i>. (Often the negative factors are forgotten.). And if the factors add up to negative 10 then they must both be negative (since you can't add two positives and get a negative). The only pairs of negative factors of 9 are: -3 and -3 or -1 and -9. Only -1 and -9 add up to -10 so you numerator factors into {{{(x-1)(x-9)}}}.<br>
Using the same type of logic on the denominator, we look for factors of -3 that add up to +2. The only pair of factors of -3 that add up to +2 are: -1 and 3. So the denominator factors into: {{{(x-1)(x+3)}}}<br>
So the factored fraction is:
{{{((x-1)(x-9))/((x-1)(x+3))}}}
We can now see that there is a common factor, (x-1), that we can cancel:
{{{(cross((x-1))(x-9))/(cross((x-1))(x+3))}}}
leaving
{{{(x-9)/(x+3)}}}
which is the simplified/reduced fraction. (Don't try to cancel an x or a 3 here. They are not factors and <i>only factors can be canceled!</i>)