【Analyze】 , 【resolve】 , 【dissect】 and 【break down】 all mean to divide a complex whole or unit into its component parts or constituent elements.

When their corresponding nouns (analysis, resolution, dissection, breakdown) denote such a division, they are similarly applied and are distinguishable by the same implications.

【Analyze】 and analysis presuppose a personal agent and stress division for the sake of determining a thing’s true nature or the inner relationship of its parts.

• 【analyze】 a sentence
• 【analyze】 the plot of a novel
• Liebig, by analyzing foodstuffs of every kind, came to the conclusion that the principal elements of food are proteins, fats, and carbohydrates
• he would take a place or a fork or a bell, set it to ringing by a blow, and 【analyze】 the combination of musical notes which it emitted
  —Darrow
• analysis of material objects into electrons and protons
  —Inge

Sometimes these words specifically suggest an intent to discover or uncover qualities, causes, effects, motives, or possibilities often as a basis for action or for a judgment.

• 【analyze】 the condition
• of a business before investing in it 【analyze】 the potential market for cotton
• I could not then so far 【analyze】 all that is roughly lumped together as "religion" as to disentangle the essential from the accidental
  —Ellis

Often, especially in chemistry and other sciences, the words imply close examination as for detecting impurities or the quantity or quality of each of the constituent elements.

• 【analyze】 a city’s water supply
• 【analyze】 a hypothesis

【Resolve】 and resolution only occasionally imply a personal agent; they therefore seldom suggest more than the actual division or separation into elements or parts.

• star clusters . . . so distant that even in telescopes of great power they could not be resolved
  —Lockyer
• nothing but death was strong enough to shatter that inherited restraint and 【resolve】 it into tenderness
  —Glasgow

Sometimes, especially in medicine, 【resolve】 or resolution implies a breaking up or disintegration and usually, as a consequence, a dissipation or scattering.

• resolution of the exudate during recovery from pneumonia

【Dissect】 and dissection stress the actual and visible separation of parts; thus, one dissects or cuts into sections an animal or a plant so that its physical structure can be studied; in extended or figurative use, one dissects something when one takes it to pieces and offers it for examination from every angle and in every detail, pleasant or unpleasant.

• we 【dissect】 the senseless body, and why not the mind?
  —Wordsworth
• the student who is willing . . . to discipline his mind by the patient correlation of facts and the fearless dissection of theories
  —Baerlein
• when you . . . 【dissect】 the Odyssey, what amazing artifice is found under that apparently straightforward tale!
  —Quiller-Couch

【Break down】 and breakdown are used chiefly with reference to topics (as financial reports, statements, or estimates) that involve numbers or quantities, or to substances separated by chemical agents.

In both situations these words imply reduction to simpler parts or divisions.

• a consolidated balance sheet is often accompanied by a breakdown giving a detailed statement for each of the main items involved

In chemistry division into simpler substances rather than into elements is typically implied.

• proteins are broken down by enzymes into amino acids