abstract algebra: a problem related to polynomial rings and ideals


I need to demonstrate the following result, but I don't know how to proceed:

Leave $ k (X) $ be a polynomial ring and leave $ I, J $ be ideal of $ k (X) $. In the ring $ k (X, t) $ ($ t $ it is a new indeterminate) is considered the ideal $ L = tI + (1-t) J $. Test it $ I cap J = L cap k (X) $.

I would be really grateful if you could help me.