Final answer:
To rewrite the given rational expressions with the common denominator (3x-8)(x-5)(x-3), we must factor the denominators of the original expressions, and multiply the numerators and denominators by any missing factors to achieve the common denominator.
Step-by-step explanation:
The student is asking to rewrite two given rational expressions with a common denominator of (3x-8)(x-5)(x-3). The given expressions are 4/3x²-23x+40 and 9x/3x²-17x+2. The process involves finding factors of the denominators and then multiplying both the numerator and the denominator of each fraction by any missing factors to obtain an equivalent expression with the common denominator of (3x-8)(x-5)(x-3).
For the first expression, 4/(3x²-23x+40), we factor the denominator and get 4/((3x-8)(x-5)). It is missing the factor (x-3), so we multiply the numerator and the denominator by (x-3). For the second expression, 9x/(3x²-17x+2), we initially factor the denominator to get 9x/((x-1)(3x-2)). However, this denominator does not match the desired denominator. Here, we would need to identify a mistake, as the factoring seems incorrect because the factors do not match the requested denominator. We would need to reevaluate the factoring or correct any potential errors.
Once the correct factors of both denominators are found, we would achieve the common denominator by multiplying each fraction by the missing factors respectively in their numerators and denominators. This allows us to rewrite both expressions as equivalent rational expressions with the common denominator (3x-8)(x-5)(x-3).