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\((x+y)^n\)
 expand

Applying the
Binomial Theorem

Master the art of expanding \((x + y)^n\) using Pascal's Triangle and binomial coefficients to unlock powerful polynomial expansions.

Algebra 2

What Is a Binomial?

A binomial is a polynomial with exactly two terms. When we raise a binomial to a positive integer power, the result is a sum of terms with specific coefficients, powers, and patterns.

Binomial
An expression of the form \((x + y)\) with two terms separated by addition or subtraction.
Expansion
Writing \((x + y)^n\) as a sum of individual terms, each with a coefficient and specific powers of \(x\) and \(y\).

The Magic of Pascal's Triangle

Pascal's Triangle is a triangular array where each entry is the sum of the two entries directly above it. Each row gives the coefficients for expanding \((x + y)^n\).

n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

Why Does This Matter?

Expanding binomials by hand (multiplying over and over) becomes impossibly tedious for large powers. The Binomial Theorem gives us a direct formula to write any term without multiplying everything out.

Probability
Binomial coefficients count the number of ways to choose outcomes in coin flips, surveys, and experiments.
Combinatorics
\(\binom{n}{k}\) counts the number of ways to choose \(k\) items from \(n\), a foundation of counting theory.
Calculus
Taylor series and approximations rely on binomial expansions to represent functions as infinite sums.

The Binomial Theorem

For any positive integer \(n\) and any numbers \(x\) and \(y\):

Each term in the expansion has three parts:

\(\binom{n}{k}\)
The binomial coefficient: how many ways to choose \(k\) items from \(n\).
\(x^{n-k}\)
Power of the first term decreases from \(n\) down to \(0\).
\(y^k\)
Power of the second term increases from \(0\) up to \(n\).

Understanding \(\binom{n}{k}\)

This is read "\(n\) choose \(k\)" and counts the number of ways to select \(k\) objects from a set of \(n\). It can also be found directly from Pascal's Triangle: the entry in row \(n\), position \(k\) (counting from 0).

1
\(\binom{5}{0} = 1\) and \(\binom{5}{5} = 1\): edges of Pascal's Triangle are always 1.
2
\(\binom{5}{2} = \frac{5!}{2!\cdot3!} = \frac{120}{2 \cdot 6} = 10\).
3
Symmetry: \(\binom{n}{k} = \binom{n}{n-k}\), so \(\binom{5}{2} = \binom{5}{3} = 10\).

Pascal's Triangle vs. Formula

You can find binomial coefficients two ways. Both give the same answer: choose whichever is faster for the problem at hand.

Pascal's Triangle Factorial Formula
Method Build rows by adding adjacent entries Compute \(\frac{n!}{k!(n-k)!}\)
Best for Small \(n\) (up to ~8) Large \(n\) or a single specific term
Speed All coefficients at once One coefficient at a time
Risk Copying errors between rows Arithmetic errors in factorials

How to Apply the Theorem

1
Identify \(x\), \(y\), and \(n\): write your expression in the form \((x + y)^n\). Watch for subtraction: \((a - b)^n = (a + (-b))^n\).
2
Find the coefficients: use Pascal's Triangle (small \(n\)) or the formula \(\binom{n}{k}\) (large \(n\) or specific term).
3
Write each term: for \(k = 0, 1, 2, \ldots, n\), the term is \(\binom{n}{k}\, x^{n-k}\, y^k\).
4
Simplify: compute powers and multiply by coefficients. Combine like terms if needed.
5
Verify: the expansion should have exactly \(n + 1\) terms, and the exponents of \(x\) and \(y\) in each term should sum to \(n\).

Example 1: Expand \((x + y)^4\)

Row 4 of Pascal's Triangle gives us the coefficients: 1, 4, 6, 4, 1.

k = 0 k = 1 k = 2 k = 3 k = 4 1 4 6 4 1
\(x^4 y^0\)
\(x^3 y^1\)
\(x^2 y^2\)
\(x^1 y^3\)
\(x^0 y^4\)
 + + + + Result:
\(x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\)

Example 2: Expand \((2a - 3)^3\)

Rewrite as \((2a + (-3))^3\). Here \(x = 2a\), \(y = -3\), and \(n = 3\). The coefficients from row 3 of Pascal's Triangle are 1, 3, 3, 1.

0
\(\binom{3}{0}(2a)^3(-3)^0 = 1 \cdot 8a^3 \cdot 1 = \) \(8a^3\)
1
\(\binom{3}{1}(2a)^2(-3)^1 = 3 \cdot 4a^2 \cdot (-3) = \) \(-36a^2\)
2
\(\binom{3}{2}(2a)^1(-3)^2 = 3 \cdot 2a \cdot 9 = \) \(54a\)
3
\(\binom{3}{3}(2a)^0(-3)^3 = 1 \cdot 1 \cdot (-27) = \) \(-27\)

Example 3: Find a Specific Term

Find the 4th term in the expansion of \((x + 2)^7\).

1
Identify values: \(n = 7\), \(k = 3\), \(x = x\), \(y = 2\).
2
Coefficient: \(\binom{7}{3} = \frac{7!}{3!\cdot4!} = \frac{5040}{6 \cdot 24} = 35\).
3
Build the term: \(35 \cdot x^{7-3} \cdot 2^3 = 35 \cdot x^4 \cdot 8 = \) \(280x^4\).

Key Patterns to Remember

Term Count
\((x+y)^n\) always has exactly \(n+1\) terms in its expansion.
Exponent Sum
In every term, the exponents of \(x\) and \(y\) add up to \(n\).
Symmetry
Coefficients are symmetric: they read the same forwards and backwards.

Visualizing Coefficient Growth

The binomial coefficients for \((x+y)^6\) form a symmetric bell shape. The middle coefficient(s) are always the largest.

Coefficient k=0 k=1 k=2 k=3 k=4 k=5 k=6 0 10 20 1 6 15 20 15 6 1

Example 4: Coefficient of \(x^3\)

Find the coefficient of \(x^3\) in the expansion of \((3x + 2)^5\).

1
Set up: We need the term where \((3x)^{5-k} \cdot 2^k\) yields \(x^3\), so \(5 - k = 3\), giving \(k = 2\).
2
Coefficient: \(\binom{5}{2} = 10\).
3
Build the term: \(10 \cdot (3x)^3 \cdot 2^2 = 10 \cdot 27x^3 \cdot 4 = 1080x^3\).

Common Mistakes to Avoid

✕ Forgetting the Negative
In \((a - b)^n\), treat the second term as \((-b)\). Every odd power of \((-b)\) produces a negative term.
✕ Ignoring Inner Coefficients
In \((2x + 3)^4\), the coefficient 2 must be raised to a power too: \((2x)^3 = 8x^3\), not \(2x^3\).
✕ Off-by-One on Term Number
The "4th term" uses \(k = 3\) (not \(k = 4\)), because indexing starts at \(k = 0\).

Summary Reference

Concept Formula / Rule Example
Binomial Theorem \(\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k\) \((a+b)^3 = a^3+3a^2b+3ab^2+b^3\)
Binomial Coefficient \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) \(\binom{6}{2} = 15\)
Term Count \(n + 1\) terms \((x+y)^5\) has 6 terms
Specific Term \((k+1)\)th term uses index \(k\) 3rd term: \(k = 2\)
Subtraction \((x-y)^n = (x+(-y))^n\) Signs alternate

Question 1 of 5

How many terms are in the expansion of \((x + y)^9\)?

Question 2 of 5

What is the value of \(\binom{8}{3}\)?

Question 3 of 5

What is the 3rd term in the expansion of \((a + b)^5\)?

Question 4 of 5

What is the coefficient of \(x^2\) in the expansion of \((x - 3)^4\)?

Question 5 of 5

Which row of Pascal's Triangle gives the coefficients for \((x + y)^6\)?