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11.9 Pascal’s Triangle.

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Presentation on theme: "11.9 Pascal’s Triangle."— Presentation transcript:

1 11.9 Pascal’s Triangle

2 Powers of Binomials (You don’t need to write, just watch)
(x + y)0 = 1 (x + y)1 = 1x1 + 1y1 (x + y)2 = 1x2 + 2xy + 1y2 (x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3 Coefficients Pattern = Pascal’s Triangle!

3 Pascal’s Triangle Coefficients 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 5 1 1
same as exponent! Coefficients 1 Row 0 Row 1 1 1 1 Row 2 2 1 1 3 3 3 1 Symmetry! 1 4 6 4 1 4 1 5 10 10 5 1 5 1 6 15 20 15 6 1 6 Each row begins & ends with 1s Other # is sum of 2 #s above it

4 Ex 1) Expand (a + b)6 1 6 15 20 15 6 1 1st term: 1a6b0 2nd term: 6a5b1
Use row 6 of Pascal’s Triangle These are your coefficients! Exponents of ‘a’ decrease from 6 to 0 Powers of a: a6, a5, a4, a3, a2, a1, a0 Exponents of ‘b’ increase from 0 to 6 Powers of b: b0, b1, b2, b3, b4, b5, b6 1st term: 1a6b0 2nd term: 6a5b1 Add to 6

5 Ex 1 cont… (a + b)6 = 1a6b0 + 6a5b1+ 15a4b2+ 20a3b3+ 15a2b a1b5+ 1a0b6 Simplify: (a + b)6 = a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b ab5 + b6 How many terms? 7 (always 1 more than exponent) Sum of the exponents (each term)? 6 (always matches the exponent)

6 Homework Pg #1-23 Odd Pg. 618 #4-6

7 Recall Pascal’s Triangle from Friday!

8 Ex 1) Find the first 4 terms in the expansion of (x – 2y)7
Row 6: Row 7: Exponents of x decrease from 7 to 0 Exponents of (– 2y) increase from 0 to 7 (x – 2y)7 = 1(x)7(-2y)0 + 7(x)6(-2y)1 + 21(x)5(-2y)2 + 35(x)4(-2y)3… (x – 2y)7 = x7 – 14x6y + 84x5y2 – 280x4y3 + …

9 Ex 2) Find the expansion of (x3 – y2)5
Row 5: (x3 –y2)5 = 1 𝑥 − 𝑦 2 0 +5 𝑥 − 𝑦 2 1 +10 𝑥 − 𝑦 2 2 +10 𝑥 − 𝑦 2 3 +5 𝑥 − 𝑦 2 4 +1 𝑥 − 𝑦 = 𝑥 15 −5 𝑥 12 𝑦 𝑥 9 𝑦 4 −10 𝑥 6 𝑦 6 +5 𝑥 3 𝑦 8 − 𝑦 10

10 Ex 3) Expand (a + b)3 New way
Ex 3) Expand (a + b)3 New way! (this will help more with tomorrow’s lesson) Write it without Pascal’s, Here’s just the variables And the exponents, no coeff. Look! Matches exponent!!

11 HW Pg. 624 Q7-9 #1-11 odd, Pg. 630 T2-T9 Now it’s Modeling Monday time!!!


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