How do you expand brackets?
Multiply each term inside the bracket by whatever is outside it. With a single term outside, that is the distributive property: 3(x + 4) = 3x + 12.
With two brackets, each term of the first bracket multiplies each term of the second. Two terms times three terms gives six products before you combine. Counting the products first is a good way to check you didn't skip one.
- Take the first term of the first bracket and multiply it by every term of the second.
- Do the same with each remaining term of the first bracket.
- Write down every product with its sign.
- Combine like terms and write the result in standard form.
Worked example: expand (x + 1)³
The answer is x³ + 3x² + 3x + 1. Write the cube as a square times one more bracket, then expand in two stages.
The coefficients 1, 3, 3, 1 are not a coincidence. They are a row of Pascal's triangle, which gives you a shortcut for any power of a binomial.
How do you multiply polynomials with more than two terms?
Use the same rule: every term times every term. To multiply polynomials like (x + 2)(x² − 3x + 4), multiply x by all three terms, then 2 by all three terms.
Lining up the two rows by power, as above, makes the combining step much easier, since each column holds one set of like terms. A box (or grid) works too: write one polynomial across the top, the other down the side, and fill in each cell. This is what a polynomial multiplication calculator does behind the scenes.
How does Pascal's triangle help with binomial expansion?
Row n of Pascal's triangle gives the coefficients of (a + b)ⁿ. Each number in the triangle is the sum of the two numbers above it.
| Power n | Pascal row | Expansion of (x + 1)ⁿ |
|---|---|---|
| 2 | 1, 2, 1 | x² + 2x + 1 |
| 3 | 1, 3, 3, 1 | x³ + 3x² + 3x + 1 |
| 4 | 1, 4, 6, 4, 1 | x⁴ + 4x³ + 6x² + 4x + 1 |
When the second term isn't 1, multiply each coefficient by the matching power of that term. For (x + 2)⁴, use 1, 4, 6, 4, 1 with powers of 2 (1, 2, 4, 8, 16): x⁴ + 8x³ + 24x² + 32x + 16. The expand calculator labels these Pascal coefficients in its steps, which makes it a handy binomial expansion calculator for powers up to 6.
Which special products are worth memorizing?
Three patterns come up again and again: the square of a sum, the square of a difference, and a sum times a difference. Knowing them lets you expand in one line and spot them later when you factor.
| Pattern | Example | Expanded |
|---|---|---|
| (a + b)² = a² + 2ab + b² | (x + 3)² | x² + 6x + 9 |
| (a − b)² = a² − 2ab + b² | (x − 4)² | x² − 8x + 16 |
| (a + b)(a − b) = a² − b² | (x + 5)(x − 5) | x² − 25 |
| (a − b)(a² + ab + b²) = a³ − b³ | (x − 2)(x² + 2x + 4) | x³ − 8 |
The pattern for a power of a difference also follows Pascal's triangle, with signs that alternate. (x − 1)⁴ expands to x⁴ − 4x³ + 6x² − 4x + 1.
How do you expand three binomials?
Multiply two of them first, then multiply that result by the third. Trying to do all three at once is where terms get lost.
For (x + 1)(x + 2)(x + 3), start with (x + 1)(x + 2) = x² + 3x + 2. Then multiply by (x + 3): x³ + 3x² + 2x + 3x² + 9x + 6, which combines to x³ + 6x² + 11x + 6.
A useful check: the constant at the end is the product of the constants, 1 × 2 × 3 = 6.
What are common mistakes when expanding?
The biggest one is squaring each term separately. (x + 3)² equals x² + 6x + 9, because the two middle products 3x and 3x add up to 6x. Writing x² + 9 drops that middle term.
Other slips: dropping a negative sign when multiplying (−2 × −3 is +6), and forgetting that a term outside the bracket multiplies all the terms inside, not just the first. Count your products before combining: a 2-term bracket times a 3-term bracket should give 6.
How do you enter an expression in the expand calculator?
Type brackets side by side for multiplication, like (2x - 1)(x + 5), and use ^ for powers, like (x - 2)^5. You don't need a times sign between brackets.
The calculator works with one variable and degrees up to 6. It won't expand expressions with two letters such as (x + y)², so substitute numbers or use the Pascal method by hand for those.