1. 11.1-11.2 Binomial Theorem & Binomial Expansion

2. Pascal’s Triangle
Pascal’s Triangle Patterns – (1) Sums of Rows: sum of elements in row n is 2^n; (2) Prime Number – If 1st
element in a row is prime (0th element is “1”), then all elements in that row are divisible by it; (3) Hockey Stick pattern; (4) Magic 11’s; (5) Fibonacci’s Sequence. Check out

3. Pascal’s Triangle with even and odd numbers colored differently:

4. Expanding Binomials… What pattern(s) do you see?.....
Patterns: (1) # of terms is 1 greater than the power; (2) exponent on “a” decreases while exponent on “b” increases; (3) sum of exponents on any terms equals the power; (4) exponent on “b” is always 1 less than the number of the term (ex.: 4th term in expansion of (a+b)^5 has exponent of 3 on “b”); and (5) coefficients come from corresponding row of Pascal’s Triangle.
What pattern(s) do you see?.....

5. Pascal's Triangle
Use this triangle to expand binomials of the form (a+b)n. Each row corresponds to a whole number n. The first row consists of the coefficients of (a+b)n when n = 0.
Example 1 1
Example 2

6. Expand (a + b)6 1 6 15 20 15 6 1 (coefficients from Pascal’s triangle)
1a6 6a5 15a4 20a3 15a2 6a1 1a (exponents of a begin with 6 and decrease)
1a6b0 6a5b1 15a4b2 20a3b3 15a2b4 6a1b5 1a0b6 (exponents of b begin with 0 and increase by 1)
a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + b6 (expansion in standard, simplified form)
Return to Triangle

7. Expand (x – 2)3 (let a = x and b = –2)
(coefficients from Pascal’s triangle)
1a3 3a2 3a1 1a0
1a3b0 3a2b1 3a1b2 1a0b3
Now substitute x for a and –2 for b:
1(x)3(–2) (x)2(–2) (x)1(–2) (x)0(–2)3
x3 – 6x2 + 12x – (expansion in standard, simplified form)
Return to Triangle

8. Factorial Notation
Read as “n factorial”

9. Finding a particular term in a Binomial Expansion:
Formula for finding a particular term in expansion of (a + b)n is:
Ex.: Find the 4th term in expansion of (a + b)9:
This is the 4th term, so value of r (b’s exponent) is 4 – 1 = 3.
This means the exponent for a is 9 – 3, or 6.
So, we have the variables of the 4th term: a6b3
Coefficient is:

10. Finding a particular term in a Binomial Expansion:
Formula for finding a particular term in expansion of (a + b)n is:
Ex.: Find the 8th term in expansion of (2x – y)12:
This is 8th term, exponent for b is 8 – 1 = 7.
This means the exponent for a is 12 – 7, or 5.
So, the variables of the 8th term: a5b7,
or (2x)5(–y)7.
Coefficient: