Final answer:
To find the L.C.M of the two expressions, factor each polynomial and take the highest power of all appearing factors. The L.C.M of 2x^3-50x and 2x^2+7x+15 is 2x(x+5)(x-5)(x+3), which when expanded, is 4x^3-100x.
Step-by-step explanation:
To Find the L.C.M of 2x^3−50x, 2x^2+7x+15, we need to factor each expression and then find the least common multiple that contains each factor the greatest number of times it occurs in any of the expressions.
First, let's factor each polynomial:
2x^3−50x can be factored by taking out the common factor of 2x:
2x(x^2−25) = 2x(x+5)(x−5)
2x^2+7x+15 factors to (2x+5)(x+3).
Next, we find the L.C.M by taking the highest power of each distinct factor that appears in any of the factorizations.
The L.C.M is therefore 2x(x+5)(x−5)(x+3).
Expanding these factors gives us:
2x(x^3−25x+x−15)
We can now write the L.C.M in its expanded form, which would be option (c): 4x^3−100x.