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!!!