For carbon cycle:
The central processing of metabolism of plants includes the usage of carbon. Plants are autotrophic thus they create their own food. They can do that if Carbon dioxide is present in the atmosphere. With the aid of light energy and water, carbon dioxide is converted to glucose. Glucose now can be used by plants and heterotrophic organisms in order to produce energy that can release carbon dioxide back in the atmosphere. Altogether they can form carbon biomass, sediments that can be deposited in soils, water systems and sea beds.
For nitrogen cycle:
Our atmosphere contains large amount of nitrogen however, it cannot be used directly by plants. There should be a third party that can convert nitrogen in a more usable form. They are called nitrogen-fixing organisms. Plants consume nitrogen in the form of nitrates which came from bacteria. From plants, it can be consumed by other organisms and be converted to proteins. Plants and animals eventually decay forming ammonia and then to nitrates again which can be denitrified to form the nitrogen gas again.
For Phosphorus cycle:
Phosphorus is an essential nutrient in plants. Soluble compounds of inorganic phosphorus can be taken up by plants which can be converted into organic molecules. Death and excretion makes the organic phosphorus be available for bacteria to convert it back to its inorganic state.