On Tuesday, we learned about The Binomial Theorem. We can determine the number of terms in a binomial by looking at the exponent (n) and adding it by one.
For example: ( x + y )^4 would have 5 terms.
= x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
1 2 3 4 5
There are patterns that work the same in every binomial expansion which are:
- The sum of the exponents in each term is equal to the power of the binomial expansion (n).
- The first x-variable will always equal to n then decreases by one in the next term. The y-variable appears on the second term then increases by one until it matches the value of n.
- The coefficients are the combinations of n that starts with nCo and ends with nCn.
(a + b)^n = nCo a^n b^0 + nC1 a^n-1 + nC2 a^n-2b^2 + ...nCn a^0b^n
- Another way to determine the coefficients in each term is by using Pascal's Triangle. To use the triangle, simply look at the number of terms the binomial expansion would have, that value would be the number of the row in the triangle. For example: ( x + y )^4 would have 5 terms so look at the 5th row in the triangle.
We can find a specific term in a binomial expansion without expanding it by using the formula:
tk+1 = nCk a^n-k b^k
The value of k = # of term - 1
Example: Find the 4th term of (x - 2y)^10
n = 10 a = x b = 2y k = 3
t4 = 10C3 * (x)^10-3 * (2y)^3
= 120x^7 * - 8y^3
= - 960x^7y^3
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