Final answer:
To multiply and simplify the expression (6x-12)/(x²-9x+18) x (7c-21)/(5x-10), we attempt to factor and cancel out common terms before multiplying. However, the first fraction cannot be simplified by factoring the quadratic, so we only simplify the second fraction and eliminate the common (x-2) term. The final simplified form is 42(c-3)/(5x²-45x+90).
Step-by-step explanation:
The question involves multiplying two algebraic fractions and simplifying the result. The first step is to multiply the numerators and the denominators together.
(6x-12)/(x²-9x+18) × (7c-21)/(5x-10)
We can attempt to simplify the individual fractions before multiplying by factoring. The numerator 6x-12 can be factored to 6(x-2), and the denominator x²-9x+18 can be factored to (x-6)(x-3), but only if the quadratic factors cleanly - in this case it happens to not factor, so we will leave it as it is. The second fraction can be simplified by factoring out 7 from the numerator, giving us 7(c-3), and factoring out 5 from the denominator, resulting in 5(x-2).
6(x-2)/(x²-9x+18) × 7(c-3)/(5(x-2))
Next, we will eliminate terms wherever possible, specifically any common factors in the numerators and denominators. Here, we see that the term (x-2) is both in one numerator and one denominator, which allows us to cancel it out. After simplification, we are left with:
6/(x²-9x+18) × 7(c-3)/5
Now, we have an expression that we can simply multiply across:
(6×7)(c-3)/(5(x²-9x+18))
Which simplifies to:
42(c-3)/(5x²-45x+90)
This is the simplest form we can achieve without further information about the values of x and c.
Lastly, always check the answer to ensure it is reasonable and that all common factors have been correctly canceled.