The Binomial Theorem
TL;DR: Ever tried multiplying out (a+b) raised to the seventh power by hand? Don't. The Binomial Theorem is the shortcut that hands you the whole expansion without all that grunt work. Each term carries a binomial coefficient, the same numbers you'd find in Pascal's triangle, and the powers of a and b in every term always add up to n. Read the pattern once and you can write out any expansion in seconds. It's one of those formulas that actually saves you time the moment you trust it.
Key takeaways:
- \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\).
- \(\binom{n}{k} = \dfrac{n!}{k!(n-k)!}\) is the number of ways to choose \(k\) items from \(n\).
- Powers of \(a\) decrease from \(n\) to \(0\); powers of \(b\) increase from \(0\) to \(n\).
- Coefficients form Pascal's triangle: row \(n\) is \(\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}\).
- There are \(n + 1\) terms in the expansion of \((a+b)^n\).
The Binomial Theorem: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Organize by degreeWrite terms from highest power to lowest power.
- Look for structureTry GCF, special products, grouping, or division depending on the expression.
- Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.
Worked examples
Combine like terms
- Group x squared terms.
- Group x terms.
- Combine each group.
Factor a difference of squares
- Recognize a squared term minus a squared term.
- Use a^2 – b^2.
- Write conjugate factors.
Try one before moving on
The Binomial Theorem: pop-up practice
The binomial theorem primarily helps to find the symbolic value of the algebraic expression of the form \((x + y)^n\).
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.
- \(\color{blue}{\left(x^2+4\right)^4}\)
- \(\color{blue}{\left(2x+3x^2\right)^5}\)
- \(\color{blue}{\left(2x+5\right)^3}\)
- \(\color{blue}{x^8+16x^6+96x^4+256x^2+256}\)
- \(\color{blue}{32x^5+240x^6+720x^7+1080x^8+810x^9+243x^{10}}\)
- \(\color{blue}{8x^3+60x^2+150x+125}\)
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.
Frequently Asked Questions
What is the binomial theorem formula?
The binomial theorem says (a + b)ⁿ is the sum of terms C(n, k) · a^(n−k) · b^k, for k = 0 up to n. Here C(n, k) = n! ÷ (k!(n − k)!). The powers of a count down from n while the powers of b count up from 0, and the expansion always has n + 1 terms.
What is the binomial theorem used for?
It lets you expand powers like (x + 3)⁶ quickly without multiplying the binomial by itself again and again. It is also used to find one specific term or coefficient, to approximate values such as 1.01¹⁰, and in probability, where binomial coefficients count the ways an outcome like 3 heads in 5 coin flips can happen.
How do you expand using the binomial theorem?
Write the coefficients from the correct row of Pascal’s triangle, lower the power of the first term each step, and raise the power of the second term. For (x + 2)³, the coefficients are 1, 3, 3, 1, so the result is x³ + 3·x²·2 + 3·x·2² + 2³ = x³ + 6x² + 12x + 8.
Is the binomial theorem the same as Pascal’s triangle?
They are closely linked but not the same thing. The binomial theorem is the rule for expanding (a + b)ⁿ, while Pascal’s triangle is a number pattern that supplies its coefficients. Row 4 of the triangle is 1, 4, 6, 4, 1, which are exactly the coefficients in (a + b)⁴.
How do you find a coefficient using the binomial theorem?
Use the general term C(n, k) · a^(n−k) · b^k and pick the k that gives the power you want. For the x² term in (x + 3)⁵, you need 5 − k = 2, so k = 3. The term is C(5, 3) · x² · 3³ = 10 · 27 · x² = 270x², so the coefficient is 270.
How do you prove the binomial theorem?
The standard proof uses mathematical induction. Check that the formula works for n = 1, then assume it holds for n and multiply (a + b)ⁿ by (a + b). Collecting like terms uses Pascal’s rule, C(n, k − 1) + C(n, k) = C(n + 1, k), which shows the formula also holds for n + 1.
Related EffortlessMath Lessons
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