Definition: The binomial theorem expands a power of a two-term sum, (a + b) to the n, into a sum of terms whose coefficients come from Pascal's triangle.
What does the theorem say?
For a whole number n, (a + b)n = sum of C(n, k) an-k bk for k = 0 to n. The coefficients C(n, k) are “n choose k” and match row n of Pascal’s triangle. The powers of a go down while the powers of b go up.
How do you expand a power?
Worked example: expand (2x – 1)4. The row for n = 4 is 1, 4, 6, 4, 1. Use a = 2x and b = -1.
- 1(2x)4 = 16x4
- 4(2x)3(-1) = -32x3
- 6(2x)2(1) = 24x2
- 4(2x)(-1) = -8x
- 1(-1)4 = 1
Result: 16x4 – 32x3 + 24x2 – 8x + 1.
What goes wrong?
Forgetting to raise the coefficient 2 to the power along with x, and losing the signs when b is negative. Mathbot puts each piece in parentheses before computing.
Example: (2x – 1)^4 = 16x^4 – 32x^3 + 24x^2 – 8x + 1