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binomial expansion πππ?
1 Answer
- 1 year agoFavorite Answer
(a + b)^n = nC0 * a^(n - 0) * b^0 + nC1 * a^(n - 1) * b^1 + nC2 * a^(n - 2) * b^2 + .... + nCn * a^(n - n) * b^n
(2 - y)^5 =>
5C0 * 2^5 * (-y)^0 + 5C1 * 2^4 * (-y)^1 + 5C2 * 2^3 * (-y)^2 + 5C3 * 2^2 * (-y)^3 + 5C4 * 2^1 * (-y)^4 + 5C5 * 2^0 * (-y)^5 =>
1 * 32 * 1 + 5 * 16 * (-y) + 10 * 8 * y^2 + 10 * 4 * (-y^3) + 5 * 2 * y^4 + 1 * 1 * (-y^5) =>
32 - 80y + 80y^2 - 40y^3 + 10y^4 - y^5
(2 - (2x - x^2))^5 =>
32 - 80 * (2x - x^2) + 80 * (2x - x^2)^2 - 40 * (2x - x^2)^3 + 10 * (2x - x^2)^4 - (2x - x^2)^5 =>
32 - 80 * x * (2 - x) + 80 * 2^2 * (x - 2)^2 - 40 * x^3 * (2 - x)^3 + 10 * x^4 * (2 - x)^4 - x^5 * (2 - x)^5
You just want the coefficient for x^2, so if we're dealing with x^3 or higher, we can ignore it
32 - 80x * (2 - x) + 80 * 4 * (x - 2)^2
32 - 160x + 80x^2 + 320 * (x^2 - 4x + 4)
Ignore everything by x^2
80x^2 + 320x^2 =>
400x^3
