The Binomial Theorem

The binomial theorem primarily helps to find the symbolic value of the algebraic expression of the form \((x + y)^n\). For additional educational resources, .

Related Topics

• How to Solve Infinite Geometric Series
• How to Solve Geometric Sequences
• How to Solve Arithmetic Sequences

A step-by-step guide to the binomial theorem

According to the binomial theorem, it is possible to expand any non-negative power of binomial \((x + y)\) into a sum of the form, \((x+y)^n=\begin{pmatrix}n\\ 0\end{pmatrix}x^n y^0+ \begin{pmatrix}n\\ 1\end{pmatrix}x^{n-1} y^1 + \begin{pmatrix}n\\ 2\end{pmatrix}x^{n-2} y^2+…+ \begin{pmatrix}n\\ n-1\end{pmatrix}x^1 y^{n-1}+ \begin{pmatrix}n\\ n\end{pmatrix}x^0 y^n\)

where \(n≥0\) is an integer and each \(\begin{pmatrix}n\\k\end{pmatrix}\) is a positive integer known as a binomial coafficient.

Note: When power is zero, the corresponding power expression is \(1\).

Using summation notation, the binomial theorem can be given as:

\(\color{blue}{(x+y)^n=\sum _{k=0}^n\: \begin{pmatrix}n\\k\end{pmatrix} x^{n-k}y^{k}}\) \(\color{blue}{= \sum _{k=0}^n\: \begin{pmatrix}n\\k\end{pmatrix} x^{k}y^{n-k}}\)

The binomial theorem formula

The binomial theorem formula helps to expand a binomial that has been increased to a certain power. The binomial theorem states: if \(x\) and \(y\) are real numbers, then for all \(n ∈ N\):

\(\color{blue}{(x+y)^n=\sum _{r=0}^n\: (^nC_r)x^{n-r}y^{r}}\)

where, \(\color{blue}{^nC_r}\)\(\color{blue}{=\frac{n!}{r!(n-r)!}}\)

Properties of the binomial theorem

• The number of coefficients in the binomial expansion of \((x+y)^n\) is equal to \((n+1)\).
• In the expansion of \((x+y)^n\),there are \((n+1)\) terms.
• The first and the last terms are \(x^n\) and \(y^n\) respectively.
• From the beginning of the expansion, the powers of \(x\) decrease from \(n\) to \(0\), and the powers of \(a\) increase from \(0\) to \(n\).
• The general term in the expansion of \((x + y)^n\) is the \((r +1)^{th}\) term that can be represented as \(T_{r+1}\), \(T_{r+1}=\)\(^nC_r\)\(x^{n-r}y^r\)
• The binomial coefficients in the expansion are arranged in an array called the Pascal triangle. This developed model can be summarized with a binomial theorem formula.
• In the binomial expansion of \((x+y)^n\), the \(r^{th}\) term from the end is \((n-r+2)^{th}\) term from the beginning.
•  If \(n\) is odd, then in \((x + y)^n\), the middle terms are \(\frac{(n+1)}{2}\) and \(\frac{(n+3)}{2}\).
• If \(n\) is even, then in \((x + y)^n\), the middle term \(=(\frac{n}{2})+1\)

The Binomial Theorem – Example 1:

Expand \((x+2)^5\) using the binomial theorem.

Use this formula to expand: \(\color{blue}{(x+y)^n=\sum _{k=0}^n\: \begin{pmatrix}n\\k\end{pmatrix} x^{n-k}y^{k}}\)

\(=\sum _{k=0}^5\: \begin{pmatrix}5\\k\end{pmatrix}x^{(5-k)}.2^k\)

\((x+2)^5=\frac{5!}{0!(5-0)!}x^5.2^0+ \frac{5!}{1!(5-1)!}x^4.2^1+ \frac{5!}{2!(5-2)!}x^3.2^2+ \frac{5!}{3!(5-3)!}x^2.2^3+ \frac{5!}{4!(5-4)!}x^1.2^4+ \frac{5!}{5!(5-5)!}x^0.2^5 \)

\(x^5+10x^4+40x^3+80x^2+80x+32\)

Exercise for the Binomial Theorem

Find binomial expansion by using the binomial theorem.

1. \(\color{blue}{\left(x^2+4\right)^4}\)
2. \(\color{blue}{\left(2x+3x^2\right)^5}\)
3. \(\color{blue}{\left(2x+5\right)^3}\)

1. \(\color{blue}{x^8+16x^6+96x^4+256x^2+256}\)
2. \(\color{blue}{32x^5+240x^6+720x^7+1080x^8+810x^9+243x^{10}}\)
3. \(\color{blue}{8x^3+60x^2+150x+125}\)

Recommended EffortlessMath Books

For a thorough Algebra II treatment of the binomial theorem and the algebra you’ll need to use it well, Algebra II for Beginners walks through every step with worked examples and practice. If you’re heading into calculus or pre-calc, Pre-Calculus for Beginners extends these ideas into power series and combinatorial identities.

Algebra II for Beginners 2026 The Ultimate Step by Step Guide to Acing Algebra II

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

Rated 5.00 out of 5 based on 1 customer rating

Satisfied 1 Students

Frequently Asked Questions

What is the binomial theorem?

The binomial theorem is a formula for expanding \((a + b)^n\) into a sum of terms, where \(n\) is a non-negative integer: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}.\] Each term has a binomial coefficient, a power of \(a\), and a power of \(b\). The exponents of \(a\) and \(b\) always add up to \(n\).

What’s a binomial coefficient?

A binomial coefficient \(\binom{n}{k}\) (“n choose k”) counts the number of ways to choose \(k\) items from \(n\) without regard to order: \(\binom{n}{k} = \dfrac{n!}{k!(n-k)!}\). For \(\binom{5}{2}\): \(\dfrac{5!}{2! \cdot 3!} = \dfrac{120}{2 \cdot 6} = 10\). Binomial coefficients appear as the entries of Pascal’s triangle.

What’s Pascal’s triangle?

Pascal’s triangle is a number triangle where each entry is the sum of the two directly above it. Row \(n\) (starting from row 0) gives the binomial coefficients \(\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}\). Row 4 is \(1, 4, 6, 4, 1\), so \((a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\).

How do I find a single term without expanding everything?

The \((k+1)\)th term of \((a+b)^n\) is \(\binom{n}{k} a^{n-k} b^{k}\). To find the term with \(x^3\) in \((x + 2)^7\): set \(n – k = 3\), so \(k = 4\). The term is \(\binom{7}{4} x^3 (2)^4 = 35 \cdot x^3 \cdot 16 = 560 x^3\). Skipping the full expansion saves enormous time for big \(n\).

How does the binomial theorem connect to probability?

The binomial probability formula \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\) is exactly the \(k\)th term of the expansion of \((p + (1-p))^n\) – which equals 1, confirming the probabilities sum to 1. The coefficients in Pascal’s triangle are literally the number of ways the successes can happen in a sequence of trials.

What if the second term has a negative sign?

Treat it as a binomial with negative \(b\). For \((a – b)^n\), write it as \((a + (-b))^n\) and let \(b\) carry its sign through. Even-power terms of \(-b\) stay positive, odd-power terms become negative, so the signs of the expansion alternate. \((x – 1)^4 = x^4 – 4x^3 + 6x^2 – 4x + 1\).

How does the binomial theorem extend to non-integer exponents?

For \(|x| < 1\), Newton's generalized binomial series gives \((1 + x)^r = \sum_{k=0}^{\infty} \binom{r}{k} x^k\) for any real \(r\), where \(\binom{r}{k} = \dfrac{r(r-1)\cdots(r-k+1)}{k!}\). The series has infinitely many terms when \(r\) isn't a non-negative integer. This shows up in calculus and physics but isn't usually on Algebra II tests.

Walk me through expanding \((x + 2)^5\)

Coefficients (row 5 of Pascal’s triangle): \(1, 5, 10, 10, 5, 1\). Powers of \(x\): \(x^5, x^4, x^3, x^2, x, 1\). Powers of 2: \(1, 2, 4, 8, 16, 32\). Multiply term by term: \(x^5 + 5 \cdot 2 \cdot x^4 + 10 \cdot 4 \cdot x^3 + 10 \cdot 8 \cdot x^2 + 5 \cdot 16 \cdot x + 32 = x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32\).

How many terms does \((a+b)^n\) have?

Exactly \(n + 1\) terms. For \((a+b)^5\), there are 6 terms; for \((a+b)^{10}\), there are 11 terms. The exponent of \(a\) runs from \(n\) down to 0, and the exponent of \(b\) runs from 0 up to \(n\), giving \(n + 1\) different combinations.

Where does the binomial theorem show up on tests?

Algebra II, pre-calculus, the SAT Math 2 Subject Test (legacy), AP Calculus (in series), AP Statistics (for binomial probability), and college math placement exams. Common question types: expand a binomial, find a specific term or coefficient, or use the theorem to simplify a probability expression.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• Conditional and binomial probabilities
• Probability mass function
• Probability of an event
• How to solve probability problems
• Expected value of random variables