A polynomial identity holds for every value: (a + b)² = a² + 2ab + b², and the middle term is what people forget. Row n of Pascal’s triangle holds the coefficients of (a + b)ⁿ, the entries are the combinations nCr, and the powers of a fall while the powers of b rise, each term totalling n. Substituting a = b = 1 checks any expansion.
Step 1: Picture It
Start with something you already know.
Four coin flips
- You flip a coin four times and write down each result, H for heads or T for tails.
- One run might read T H H T. Another might read H H T T.
- Each flip doubles the number of possible runs, so there are 16 in all.
- Now sort the 16 runs by how many heads each one has.
Expanding (a + b)⁴ is four flips: from each of the four brackets you take a (call it tails) or b (call it heads).
The coefficient of a²b² counts the runs with exactly two heads: the number of ways to choose which 2 of the 4 brackets give a b.
Where the picture stops: a coin cannot carry a number or a minus sign inside a bracket, as in (2x − 3)⁴. Those get multiplied in afterward, as the powers of a and b.
The 16 runs of four flips, counted by their number of heads.
| Heads in four flips | Number of runs |
|---|---|
| 0 | 1 |
| 1 | 4 |
| 2 | 6 |
| 3 | 4 |
| 4 | 1 |
That is row 4 of Pascal’s triangle. The counts add to 16, which is 2⁴, the same total you get by putting a = b = 1 into (a + b)⁴.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
You know (x + 1)² = x² + 2x + 1. Multiply that by one more (x + 1) to get (x + 1)³. Type the result with no brackets.
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Cubes
a³ − b³ = (a − b)(a² + ab + b²) and a³ + b³ = (a + b)(a² − ab + b²). x³ − 27 = (x − 3)(x² + 3x + 9). Check by expanding: x³ + 3x² + 9x − 3x² − 9x − 27 = x³ − 27.
The triangle
1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1; 1 5 10 10 5 1. Each row starts and ends with 1, and every other entry is the sum of the two above. Row 5 is ready for a fifth power.
They are combinations
The entry 6 in row 4 is 4C2 = 4!/(2! 2!) = 6: the number of ways to choose which 2 of the 4 brackets contribute a b. Row 6, position 2: 6C2 = 15, without writing the triangle.
With coefficients inside
(2x − 3)⁴: a = 2x, b = −3. The a-powers are 16x⁴, 8x³, 4x², 2x, 1; the b-powers 1, −3, 9, −27, 81. Multiply each pair by its row-4 coefficient: 16x⁴ − 96x³ + 216x² − 216x + 81.
One term only
The x³ term of (x + 3)⁶ is 6C3 · x³ · 3³ = 20 · 27 x³ = 540x³. The term with bʳ is nCr · aⁿ⁻ʳ · bʳ; no need to expand the other six.
Check with x = 1
(2 − 3)⁴ = 1, and 16 − 96 + 216 − 216 + 81 = 1. One substitution catches a wrong sign or a dropped coefficient anywhere in the expansion.
Step 3: Try It Yourself
Tap and try it out.
Expand (2x − 3)⁴ and check it; then find the x³ term of (x + 3)⁶ without expanding
- 1row 4: 1, 4, 6, 4, 1; a = 2x, b = −3Pascal, and name the parts
Step 0 of 4
4C0
4C1
4C2
4C3
4C2 has the most. It has 5 more than 4C0.
Step 4: In Real Life
Ten years of growth
(1 + r)¹⁰ is what a decade of growth multiplies by. The binomial theorem expands it: 1 + 10r + 45r² + …, a quick estimate of ten years’ interest without a calculator.
Step 5: Watch an Example
One step at a time.
Watch Sana Expand (x + 2)⁵
A fresh power, built from Pascal’s triangle in four moves and checked at the end with x = 1.
- Step 1
The power is 5, so Sana reads row 5 of the triangle: 1, 5, 10, 10, 5, 1. Six numbers, so the answer will have six terms. Here a = x and b = 2.
Watch Leo Expand (2x − 1)³
A number and a minus sign inside the bracket. Leo sets up the two lists, and you finish the terms.
- Step 1
The power is 3, so the row is 1, 3, 3, 1. Leo names the parts with their signs: a = 2x and b = −1. The minus belongs to b and travels with it.
Watch Ana Factor x³ + 8
Not a power to expand this time but a sum of two cubes to factor, using the cube identity.
- Step 1
Ana spots that 8 is a cube: 8 = 2³. So x³ + 8 is x³ + 2³, a sum of two cubes with a = x and b = 2.
Step 6: Your Turn
Practice makes it stick.
A Decade at 3%
Problem 1 of 15
$1,000 earns 3% a year for 10 years, so it is multiplied by (1 + 0.03)¹⁰. Row 10 of Pascal’s triangle begins 1, 10, 45. Using only the first three terms of the expansion, 1 + 10r + 45r², estimate what the $1,000 grows to, in dollars, to the nearest cent.
dollars
The Bigger Box
Problem 2 of 15
A cube-shaped storage box has edges of 10 inches, so it holds 10³ = 1,000 cubic inches. The larger model has edges of 11 inches. Writing 11 as 10 + 1 and expanding (10 + 1)³, how many more cubic inches does the larger box hold?
cubic inches
Expand and Pick Out
Row first, then the two lists of powers, then multiply. Check with x = 1 when you can.
Problem 3 of 15
Expand (x + 3)². Type it with no brackets.
Problem 4 of 15
What is 5C2?
Problem 5 of 15
Expand (x − 2)³. Type it with no brackets.
Problem 6 of 15
Without expanding the rest, what is the coefficient of x² in (x + 3)⁴?
Problem 7 of 15
x³ − 64 factors as (x − 4) times a quadratic. What is the quadratic’s constant term?
Whose Step?
One line in each page is a slip. Find it, then fix it.
Problem 8 of 15
Dev expanded (x − 2)⁴. One line is a slip. Which?
Dev’s expansion of (x − 2)⁴
- 1row 4: 1, 4, 6, 4, 1
- 2x-powers: x⁴, x³, x², x, 1
- 3b-powers: 1, 2, 4, 8, 16
- Answerx⁴ + 8x³ + 24x² + 32x + 16
Problem 9 of 15
In the correct expansion of (x − 2)⁴, what is the coefficient of x³?
Problem 10 of 15
Ana found the x³ term of (x + 2)⁶ without expanding the rest. One line is a slip. Which?
Ana’s x³ term of (x + 2)⁶
- 1x⁶⁻ʳ = x³, so r = 3
- 26C3 = 6! / (3! 3!) = 20
- 3bʳ = 2¹ = 2
- Answer20 · 2 · x³ = 40x³
Problem 11 of 15
What is the coefficient of x³ in (x + 2)⁶, correctly?
Pick the Move
Decide how to start or how to check. No need to finish the expansion.
Problem 12 of 15
You want only the x² term of (x + 3)⁷, using nCr · aⁿ⁻ʳ · bʳ. Which r do you use?
Problem 13 of 15
Maya expanded (3x − 2)⁴ and wants to know if she dropped a minus sign somewhere. Which single check can catch that?
Same Thing?
An identity holds for every value of the letters. Judge which of these do.
Problem 14 of 15
Select every expression that equals (a + b)² for all values of a and b.
Problem 15 of 15
Select every statement that is a true identity.
Step 7: Quick Check
Show what you know.
Question 1 of 2
How many terms does (a + b)⁸ have?
Question 2 of 2
Where do the binomial coefficients come from?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
For which number k does the expansion of (x + k)⁴ have an x³ term of 20x³?
Stretch 2 of 2
Which row of Pascal’s triangle has entries that add up to 256?
What You Learned
- Row n of Pascal’s triangle gives the coefficients of (a + b)ⁿ, and its entries are the combinations nCr.
- The powers of a fall while the powers of b rise, always totalling n.
- With coefficients or signs inside, list the a-powers and b-powers first, then multiply.
- The term with bʳ is nCr · aⁿ⁻ʳ · bʳ, and x = 1 checks the whole expansion.
For the grown-up
The misconception this lesson targets is treating a power of a sum as the sum of the powers: (a + b)² = a² + b², and its cousin a³ + b³ = (a + b)³. Both feel natural because a power does distribute over a product, (ab)² = a²b², just not over a sum. The second slip to watch is a sign left behind: in (x − 2)⁴ the b is −2, and a student who writes b = 2 gets every other term wrong. Putting x = 1 into both the bracket and the expansion catches either one in a single line.