Binomial theorem is one of 3 polynomials formulas in the algebra section of this library, and it is used at high school · ap level.
Why binomial theorem works
Expanding means choosing either x or y from each of the n brackets and multiplying the choices. Any term with k copies of y arises once for every way of picking which k brackets supplied them, and that count is exactly the binomial coefficient in front.
What each symbol means
$n$ is the power and $k$ indexes each term.
Binomial theorem: when it holds
$n$ must be a nonnegative integer for this finite form.
When it stops applying
The finite sum shown needs n to be a whole number that is zero or greater. For a fractional or negative exponent the expansion never terminates; it becomes an infinite series that only converges when the second term is smaller than the first in size.
Binomial theorem: a worked example
$(x+y)^3=x^3+3x^2y+3xy^2+y^3$.
The mistake to avoid
What people do: Distributing the exponent across the sum, writing (x + y) cubed as x cubed plus y cubed.
Why it goes wrong: Substituting x = y = 1 shows the gap at once: the real value is 2 cubed, which is 8, while the shortcut gives 2. The cross terms carry most of the total.
Do this instead: Expand the brackets one at a time for a small power, or read the coefficients off the matching row of Pascal's triangle.
Binomial theorem: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $n$ is the power and $k$ indexes each term.
- Check the conditions before substituting. $n$ must be a nonnegative integer for this finite form.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Polynomials
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where binomial theorem comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Factoring Polynomials — the lesson behind this formula: reverse multiplication to expose roots and structure.
- Degree Of Polynomial Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about binomial theorem
How do I get one term without expanding the whole thing?
Use a single piece of the sum with the k you want. The coefficient of x squared y cubed in the fifth power is the number of ways to choose 3 brackets out of 5, which is 10.
Where does Pascal's triangle fit in?
Each row of the triangle lists the coefficients for one power, because every entry is the sum of the two above it, matching how the choices split at each new bracket.
What changes when the binomial is something like 2x - 3?
Substitute the whole pieces, brackets and all. In the fourth power, the x cubed term is 4 times (2x) cubed times -3, which works out to -96x cubed.
Do the coefficients in a row always add to a power of 2?
Yes, because setting both variables to 1 makes the left side 2 to the n. Row five adds to 1 + 5 + 10 + 10 + 5 + 1, which is 32.