what are the key features of x/(x^2-3x+2)?
To determine the key features of the expression x/(x^2-3x+2), we can analyze its factors, zeros, asymptotes, and domain.
1. Factors:
The expression x/(x^2-3x+2) can be written as x/((x-1)(x-2)). The numerator, x, is a linear factor, while the denominator has two quadratic factors.
2. Zeros:
To find the zeros of the expression, we need to set the numerator equal to zero. Therefore, x = 0. The denominator, (x-1)(x-2), does not affect the zero of the expression, as long as it is not divided by zero.
3. Vertical Asymptotes:
Vertical asymptotes occur when the denominator is equal to zero, causing the expression to be undefined. So, we set the denominator (x-1)(x-2) equal to zero and solve for x. We get x = 1 and x = 2. Hence, there will be vertical asymptotes at x = 1 and x = 2.
4. Domain:
The domain refers to the set of all possible input values for the expression. In this case, as long as the denominator (x-1)(x-2) is not equal to zero, the expression is defined. Therefore, the domain is all real numbers except x = 1 and x = 2.
Overall, the key features of the expression x/(x^2-3x+2) include:
- Zeros: x = 0
- Vertical Asymptotes: x = 1, x = 2
- Domain: All real numbers except x = 1 and x = 2