Final answer:
To simplify the rational expression (3x²-17x-28)/(x²-10x+21), factor both the numerator and the denominator, then cancel common terms. The simplified result is (3x+4)/(x-3).
Step-by-step explanation:
To simplify the rational expression (3x²-17x-28)/(x²-10x+21), we need to factor both the numerator and the denominator before canceling out any common factors. First, let's find factors of the numerator 3x²-17x-28 and the denominator x²-10x+21.
For the numerator, factors of -84 (3 times -28) that add up to -17 are -21 and 4. So it can be factored as (3x+4)(x-7). For the denominator, factors of +21 that add up to -10 are -7 and -3. It can be factored as (x-7)(x-3). Now the expression looks like:
((3x+4)(x-7))/((x-7)(x-3))
You can see that the (x-7) term is present in both the numerator and the denominator. We can eliminate these common factors to simplify the expression to:
(3x+4)/(x-3)
After simplification, make sure to check the answer to ensure that it is reasonable. Since there is no further simplification possible, and no extraneous solutions introduced, this is our final simplified rational expression.