Binomial Theorem
Expand (a + b)ⁿ using Pascal's triangle coefficients.
About this topic
The binomial theorem gives the expansion of (a + b)ⁿ without multiplying repeatedly: the coefficients come from Pascal's triangle, or the formula C(n, k), and the powers of a decrease while those of b increase. For example (x + 2)³ = x³ + 6x² + 12x + 8.
The kth term is C(n, k) aⁿ⁻ᵏ bᵏ. The theorem is closely tied to combinations and probability, and it is used in algebra, statistics and approximations.
Worked example: expand (2x − 1)⁴. The coefficients from Pascal's row 1, 4, 6, 4, 1 give (2x)⁴ − 4(2x)³ + 6(2x)²(1) − 4(2x) + 1 = 16x⁴ − 32x³ + 24x² − 8x + 1. Notice that the negative sign alternates the signs of the terms. For a single term, the 3rd term of (x + y)⁵ is C(5, 2)x³y² = 10x³y².
The theorem is used in probability, where the chance of exactly k successes in n tries includes C(n, k), and in estimating numbers such as 1.01¹⁰ ≈ 1 + 0.1. A common error is forgetting that coefficients and powers apply to the entire term, so (2x)³ is 8x³, not 2x³. Another is mixing up the term number and k: the kth term uses exponent k − 1. Also, C(n, 0) = C(n, n) = 1, and the row sums double each time.
Key ideas
- Coefficients come from row n of Pascal's triangle
- Powers of a fall, powers of b rise
- Total coefficients sum to 2ⁿ
Quick check
Use cases: where Binomial Theorem helps
- Homework: review the key idea behind Binomial Theorem before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Binomial Theorem without notes.
- Visual learning: play with the interactive lab to see how Binomial Theorem changes when you adjust its inputs.
- Teaching a friend: explain Binomial Theorem in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Binomial Theorem?
Expand (a + b)ⁿ using Pascal's triangle coefficients.
What will I learn in Binomial Theorem?
Coefficients come from row n of Pascal's triangle. Powers of a fall, powers of b rise. Total coefficients sum to 2ⁿ.
Is Binomial Theorem part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Binomial Theorem lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Binomial Theorem?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.