Binomial Theorem Expand
If you multiply (x + y) by itself 2 times, you get x^2 + 2xy + y^2. But what if you had to multiply (x + y) by itself 10 times? Writing that out by hand would take forever — unless you know the pattern hiding inside every expansion, a pattern first proven by Isaac Newton in 1665 and known today as the Binomial Theorem.
What You'll Learn
- What a binomial is and what 'expanding' it means - How Pascal's Triangle gives you the numbers instantly, without multiplying - How to expand (x + y)^3 and (x + y)^4 step by step - Why the pattern always adds up to matching powers
What Does 'Expand' Mean?
A binomial is just an expression with two terms, like (x + y). To 'expand' (x + y)^n means writing out the full sum without the exponent — turning (x+y)^2 into x^2 + 2xy + y^2. Every term in the expansion follows a strict rule: the powers of x go DOWN while the powers of y go UP, and every term's powers add up to n.
Pascal's Triangle Gives You the Coefficients
Instead of multiplying binomials by hand, you can read the coefficients straight from Pascal's Triangle. Row 0 is just 1. Row 1 is 1, 1. Row 2 is 1, 2, 1 (matching x^2 + 2xy + y^2). Row 3 is 1, 3, 3, 1. Row 4 is 1, 4, 6, 4, 1. Each number is the sum of the two numbers directly above it. So for (x+y)^3, you take row 3's numbers (1, 3, 3, 1) and attach them to terms with matching powers: 1x^3 + 3x^2y + 3xy^2 + 1y^3.
Worked Example: Expanding (x + y)^4
Step 1: Find row 4 of Pascal's Triangle: 1, 4, 6, 4, 1. Step 2: Start x's power at 4 and count down to 0: x^4, x^3, x^2, x^1, x^0. Step 3: Start y's power at 0 and count up to 4: y^0, y^1, y^2, y^3, y^4. Step 4: Multiply each coefficient by its matching x-power and y-power term: 1x^4y^0 + 4x^3y^1 + 6x^2y^2 + 4x^1y^3 + 1x^0y^4 Step 5: Clean it up (drop y^0 and x^0 since anything to the power 0 is 1): x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4.
Every expanded binomial should have exactly n+1 terms. (x+y)^4 has 5 terms — count them above and you'll see it matches.
What are the coefficients when you expand (x+y)^3 using Pascal's Triangle?
In the expansion of (x+y)^5, what must be true about the powers of x and y in every single term?
Expand and Verify
Using Pascal's Triangle, expand (x+y)^5 fully by hand, showing all 6 terms with correct coefficients and powers. Then verify your answer by substituting x=1 and y=1 into both the original (x+y)^5 = 2^5 = 32 and your expanded answer — the sum of all your coefficients should equal 32. Submit your full expansion plus the verification step.
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