science

Trinomial expansion

Trinomial expansion is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Trinomial expansion rather than just read about it. In short: In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by ( a + b + c ) n = ∑ i , j , k i + j + k = n ( n i , j , k ) a i b j c k , {\displaystyle (a+b+c)^{n}=\sum _{{i,j,k} \atop {i+j+k=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}\;\!c^{k},} where n is a nonnegative integer and the sum is taken over all combinations of nonnegative indices i, j, and…

Key takeaways

Trinomial expansion belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Trinomial expansion to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Trinomial expansion from memory before moving on to harder problems.

Reference excerpt

In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by

( a + b + c ) n = ∑ i , j , k i + j + k = n ( n i , j , k ) a i b j c k , {\displaystyle (a+b+c)^{n}=\sum _{{i,j,k} \atop {i+j+k=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}\;\!c^{k},}

where n is a nonnegative integer and the sum is taken over all combinations of nonnegative indices i, j, and k such that i + j + k = n. The trinomial coefficients are given by

( n i , j , k ) = n ! i ! j ! k ! . {\displaystyle {n \choose i,j,k}={\frac {n!}{i!\,j!\,k!}}\,.}

This formula is a special case of the multinomial formula for m = 3. The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.

Derivation The trinomial expansion can be calculated by applying the binomial expansion twice, setting d = b + c {\displaystyle d=b+c} , which leads to

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Trinomial expansion

Start with the simplest possible case. Write down what Trinomial expansion claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Trinomial expansion before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Trinomial expansion ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Trinomial expansion

In research: Trinomial expansion appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Trinomial expansion in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Trinomial expansion is common in secondary-school and first-year university syllabi. It links to neighbouring topics Factorial and binomial topics, so understanding it makes those chapters shorter.
In everyday life: Look for Trinomial expansion outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Trinomial expansion in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Trinomial expansion means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Trinomial expansion out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Trinomial expansion in simple terms?

In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by ( a + b + c ) n = ∑ i , j , k i + j + k = n ( n i , j , k ) a i b j c k , {\displaystyle (a+b+c)^{n}=\sum _{{i,j,k} \atop {i+j+k=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}…

Why does Trinomial expansion matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Trinomial expansion?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Trinomial expansion.