Final answer:
The given rational expression 3x^2 - 17x - 28 / x^2 - 10x + 21 simplifies to (3x + 4) / (x - 3) after factoring and canceling the common term (x - 7).
Step-by-step explanation:
Simplify the rational expression
3x^2 - 17x - 28 / x^2 - 10x + 21:
To simplify the given rational expression, we factor both the numerator and the denominator. We look for two numbers that multiply to give the product of the coefficient of the x^2 term and the constant term, and sum to the coefficient of the x term for both the numerator and the denominator.
In the numerator, 3x^2 - 17x - 28, we want two numbers that multiply to (3)(-28) = -84 and add to -17. These numbers are -21 and 4, so we can factor the numerator as:
(3x + 4)(x - 7)
For the denominator, x^2 - 10x + 21, we want two numbers that multiply to (1)(21) = 21 and add to -10. These are -3 and -7, so we can factor the denominator as:
(x - 3)(x - 7)
After factoring, the expression simplifies to:
(3x + 4) / (x - 3)
since we have canceled the common factors of (x - 7).