The Binomial Theorem 8.5.

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Presentation transcript:

The Binomial Theorem 8.5

Binomials like (x + y)4 The coefficients of a binomial expansion are called binomial coefficients. Use the Binomial Theorem to find them.

The Binomial Theorem In the expansion of (x + y)n (x + y)n = xn + nxn-1y + …+ nCrxn-ry + … + nxyn-1 + yn The coefficient of xn-ryr is nCr = The symbol is often used in place of nCr to denote binomial coefficients.

Example 1: Finding Binomial Coefficients Find each binomial coefficient. 8C2 7C0 20C15

Example 2: Finding Binomial Coefficients Find each binomial coefficient. 7C3 7C4 12C1 12C11

Pascal’s Triangle From the FRENCH mathematician Blaise Pascal (starts with the first row called Zero row) 1 1 2 1 3 3 1 1 4 6 4 1

Example 3: Using Pascal’s Triangle Use Pascal’s Triangle to find the binomial coefficients: 8C0, 8C1, 3C2, 8C3, 8C4, 8C5, 8C6, 8C7, 8C8

Example 4: Expanding a Binomial Write the expansion for the expression (x + 1)3. Write the expansion for the expression (x + 2)4.

What about (x – 1)3? To expand binomials with differences (subtraction) alternate the signs from negative to positive.

Example 5: Expanding Binomial Expressions Write the expansion for each expression. (2x – 3)4 (x – 2y)3