Use the Binomial Theorem 2.4 Use the Binomial Theorem Vocabulary A way to find coefficients in binomial expansions (a + b)2 where n is a positive integer. Pascal’s triangle n = 0 (0th row) 1 1 1 1 ___ 1 1 ___ ___ 1 1 ___ ___ ___ 1 n = 1 (1st row) n = 2 (2nd row) n = 3 (3rd row) n = 4 (4th row) The first and last numbers in each row are ___. Beginning with the second row, every other number is formed by ________ the two numbers immediately above the number. 1 adding

Use the Binomial Theorem 2.4 Use the Binomial Theorem Vocabulary Binomial expansion

Use the Binomial Theorem 2.4 Use the Binomial Theorem Example 1 Use Pascal’s triangle Use the fourth row of Pascal’s triangle to find the numbers in the fifth and sixth rows of Pascal’s triangle. Solution 1 4 6 4 1 1 ___ ___ ___ ___ 1 1 ___ ___ ___ ___ ___ 1 n = 4 (4th row) n = 5 (5th row) n = 6 (6th row)

Use the Binomial Theorem 2.4 Use the Binomial Theorem Checkpoint. Complete the following exercises. Find the numbers in the eighth row of Pascal’s triangle. 1 6 15 20 15 6 1 1 ___ ___ ___ ___ ___ ___ 1 1 ___ ___ ___ ___ ___ ___ ___ 1

Use the Binomial Theorem 2.4 Use the Binomial Theorem Example 2 Expand a power of a binomial sum Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion of (x + 5)4. Solution The binomial coefficients from the fourth row of Pascal's triangle are ____, ____. ____, ____, and ____. So the expansion is as follows.

Use the Binomial Theorem 2.4 Use the Binomial Theorem Checkpoint. Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.

Use the Binomial Theorem 2.4 Use the Binomial Theorem Checkpoint. Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.

Use the Binomial Theorem 2.4 Use the Binomial Theorem Example 3 Expand a power of a binomial difference Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion of (x - 6)3. Solution The binomial coefficients from the third row of Pascal's triangle are ____, ____. ____, and ____. So the expansion is as follows.

Use the Binomial Theorem 2.4 Use the Binomial Theorem Checkpoint. Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.

Use the Binomial Theorem 2.4 Use the Binomial Theorem Checkpoint. Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.

Use the Binomial Theorem 2.4 Use the Binomial Theorem Example 4 Expand a power of a binomial sum Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion of (5 + 2x)4. Solution The binomial coefficients from the fourth row of Pascal's triangle are ____, ____. ____, ____, and ____. So the expansion is as follows.

Use the Binomial Theorem 2.4 Use the Binomial Theorem Checkpoint. Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.

Use the Binomial Theorem 2.4 Use the Binomial Theorem Example 5 Find a coefficient in an expansion Find the coefficient of x3 in (4x + 3)4. Solution The binomial coefficients from the fourth row of Pascal's triangle are ____, ____. ____, ____, and ____. So the expansion is as follows. The coefficients of the x3–term is (___)(___)3(___) = _____.

Use the Binomial Theorem 2.4 Use the Binomial Theorem Checkpoint. Complete the following exercise. Find the coefficient of x2 in the expansion of (7 - x)5.

Use the Binomial Theorem 2.4 Use the Binomial Theorem Pg. 78, 2.4 #1 – 22