Red pen, highlighter, GP notebook, calculator U11D10 Have out: Bellwork: Answer the following. 1) P(3 H in 5 tosses) = 2) P(2 H in 9 tosses) = +1 +1 +1 +1 +1 +1 3) P(8 H in 10 tosses) = 4) P(at least 10 H in 12 tosses) = +3 +1 +1 +1 +1 +1 total:

Using Pascal’s Triangle for Binomial Expansion Build a Pascal’s Triangle up to row six. FYI: The “1” at the top is called the ___ row. row 0 1 row 1 1 1 row 2 1 2 1 row 3 1 3 3 1 row 4 1 4 6 4 1 row 5 1 5 10 10 5 1 row 6 1 6 15 20 15 6 1 Yesterday we discovered that the entries in the triangle represent different ____________________, Combinations

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) nCr Let’s use another notation for _______________. combination nCr = Rewrite Pascal’s Triangle using notation up to row 3. ( ) ( ) ( ) 1 1 ( ) ( ) ( ) 2 2 1 2 ( ) ( ) ( ) ( ) 3 3 1 3 2 3

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) symmetry Due to the ___________ of the triangle, we can flip it around. ( ) ( ) ( ) 1 1 ( ) ( ) ( ) 2 2 1 2 ( ) ( ) ( ) ( ) 3 3 2 3 1 3 This is just another way of writing the combination notation. We will apply this notation on day 11.

...and now for something completely different... Binomial Expansion Example #1: The binomial we will expand first is (x + 1) 1 = ____ This is the _____ of the Pascal’s Triangle, called row _____. Top = _____ + _____ The ___________ of the “expanded” binomial are row ___ of Pascal’s . 1x 1 coefficients 1 1(x)2 2(x) 1 The ___________ of the expanded binomial are row ___ of Pascal’s . coefficients = _____ + _____ + _____ x + 1 2 x2 x 1 x +1

= = ( ) x2 + 2x + 1 x2 + 2x + 1 = _____ + _____ + _____ + _____ x x3 2x2 x x2 2x 1 1(x3) 3(x)2 3(x) 1 +1 coefficients 3 The ____________ of the expanded binomial are row ___ of Pascal’s .

NOTE: The powers of x match ____ in for each entry. r Without expanding the binomial the long way, use Pascal’s  to expand the following. NOTE: The powers of x match ____ in for each entry. r 1 (x + 1)4 = ______ + ______ + ______ + ______ + ______ x4 4 x3 6 x2 4 x1 1 x0 Step 1: list the coefficients of row 4 Similarly, expand: Step 2: Write the powers in descending order, starting with 4 (x + 1)5 = _____ + _____ + _____ + _____ + _____ + _____ 1 x5 5 x4 10 x3 10 x2 5 x1 1 x0 (x + 1)6 = _____ + _____ + _____ + _____ + _____ + _____ + _____ 1 x6 6 x5 15 x4 20 x3 15 x2 6 x1 1 x0 Make sure to start writing the numbers on the left side of the blanks so that you will have plenty of room!

Example #2: Now let’s expand (x + 3) (x + 3)0 = ____ (x + 3)0 = ____ This is row ___ of the Pascal’s . 1 x1 30 x0 (x + 3)1 = _____ + _____ This is row _____ of the Pascal’s . 1 1 31 1 = x + 3 (x + 3)2 = _____ + _____ + _____ 1 x2 30 2 x1 31 1 x0 32 2 This is row ____ of the Pascal’s . = x2 + 6x +9 x + 3 Step 1: list the coefficients of row 2 x + 3 x2 3x 9 Step 2: Write the powers of x in descending order, starting with 2. Step 3: Write the powers of 3 in ascending order, starting with 0.

1 3 3 1 = ______ + ______ + ______ + ______ 1. coefficients row 3 1 1 1 1 2 1 Row 3 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

= ______ + ______ + ______ + ______ 1 x3 30 3 x2 31 3 x1 32 1 x0 33 = ______ + ______ + ______ + ______ = x3 + 9x2 + 27x + 27 1. coefficients row 3 2. powers start with 3 3. powers start with 0 = _______ + _______ + _______ + _______ + _______

1 1. coefficients row 4 1 1 2. powers start with 4 1 2 1 3 3 1 1 4 6 4 1 Row 4 1 5 10 10 5 1 1 6 15 20 15 6 1 1 x4 30 4 x3 31 6 x2 32 4 x1 33 1 x0 34 = _______ + _______ + _______ + _______ + _______ = x4 + 12x3 + 54x2 + 108x + 81

Conclusion: As we move from left to right, the powers of x __________, while the powers of 3 ___________. decrease increase The ____ of the exponents of any term of (x + 3)n is ____. sum n

= _______ + ________ + _______ x2 (-2)0 x1 (-2)1 x0 (-2)2 = ______ + ______ 1 3 3 1 = x - 2 1 4 6 4 1 = -2 = _______ + ________ + _______ 1 x2 (-2)0 2 x1 (-2)1 1 x0 (-2)2 = x2 – 4x + 4

= _______ + _______ + _______ + _______ 1 x3 (-2)0 3 x2 (-2)1 3 x1 (-2)2 1 x0 (-2)3 = x3 – 6x2 + 12x – 8 -2 = 1 x4 (-2)0 4 x3 (-2)1 6 x2 (-2)2 4 x1 (-2)3 1 x0 (-2)4 = ________ + ________ + ________ + _______ + _______ = x4 – 8x3 + 24x2 – 32x + 16

=__________ + _____________ = ___ 1 1 (2x)1 (3)0 1 (2x)0 (3)1 =__________ + _____________ = 2x + 3 = _______ + ________ + _______ 1 (2x)2 (3)0 2 (2x)1 (3)1 1 (2x)0 (3)2 = 4x2 + 12x + 9

Remember to apply the exponent to the entire term, FOOL! = _________ + __________ + __________ + _________ 1 (2x)3 (3)0 3 (2x)2 (3)1 3 (2x)1 (3)2 1 (2x)0 (3)3 = 8x3 + 36x2 + 54x + 27 1 (2x)4 (3)0 4 (2x)3 (3)1 6 (2x)2 (3)2 4 (2x)1 (3)3 1 (2x)0 (3)4 = ________ + ________ + ________ + ________ + _______ = 16x4 +96x3 +216x2 +216x +81 Remember to apply the exponent to the entire term, FOOL!

Example #5: Expand using Pascal’s . You will run out of room unless you start on the far left of each blank! 1 (3x)4 (-2)0 4 (3x)3 (-2)1 6 (3x)2 (-2)2 4 (3x)1 (-2)3 1 (3x)0 (-2)4 = ___________ + __________ + __________ + _________ + _________ = 81x4 - 216x3 + 216x2 - 96x + 16

ROW 3: 1,3,3,1 1 (3x)3 (5)0 + 3 (3x)2 (5)1 + 3 (3x)1 (5)2 + 1 (3x)0 Practice: Completely expand and simplify each binomial using Pascal’s . 1) ROW 3: 1,3,3,1 1 (3x)3 (5)0 + 3 (3x)2 (5)1 + 3 (3x)1 (5)2 + 1 (3x)0 (5)3 27x3 + 135x2 + 225x + 125 2) ROW 5: 1,5,10,10,5,1 1 (2x)5 (-1)0 + 5 (2x)4 (-1)1 + 10 (2x)3 (-1)2 + 10 (2x)2 (-1)3 + 5 (2x)1 (-1)4 + 1 (2x)0 (-1)5 32x5 – 80x4 + 80x3 – 40x2 + 10x – 1 3) ROW 4: 1,4,6,4,1 1 (x)4 (-4)0 + 4 (x)3 (-4)1 + 6 (x)2 (-4)2 + 4 (x)1 (-4)3 + 1 (x)0 (-4)4 x4 – 16x3 + 96x2 – 256x + 256

4) ROW 3: 1,3,3,1 1 (3x)3 (-4)0 + 3 (3x)2 (-4)1 + 3 (3x)1 (-4)2 + 1 (-4)3 27x3 – 108x2 + 144x – 64 5) ROW 5: 1,5,10,10,5,1 1 (2x)5 (-3y)0 + 5 (2x)4 (-3y)1 + 10 (2x)3 (-3y)2 + 10 (2x)2 (-3y)3 + 5 (2x)1 (-3y)4 + 1 (2x)0 (-3y)5 32x5 – 240x4y + 720x3y2 – 1080x2y3 + 810xy4 – 243y5

Complete the Worksheet. Don’t be this guy.