P.o.D. – Find the next 3 terms
1.) −2,
−2 −2
3
,
9
,…
2.) 5x+7, -10x-14, 20x+28, …
3.) Find the 10th term in the
sequence √3, 3, 3√3, …
4.) Write a sequence that has
two geometric means between
6.4 and 12.5
5.) Find the sum of the first six
terms of the geometric series
2-8+32-128+…
6.) If r=-2 and 𝑎8 = −384, what is
the first term of the geometric
sequence?
1.)
−2 −2 −2
,
,
27 81 243
2.) -40x-56, 80x+112, -160x-224
3.) 243
4.) 6.4, 8, 10, 12.5
5.) -1638
6.) 3
9.5 – The Binomial Theorem
Learning Target: I will be able to
use the binomial theorem to
calculate binomial coefficient;
use Pascal’s triangle to calculate
binomial coefficients.
Combination – a selection of
objects where the order is not
important. The number of n
objects taken r at a time is
denoted by C(n,r) or 𝑛𝐶𝑟
𝑛!
𝐶 (𝑛, 𝑟) =
[(𝑛 − 𝑟)! 𝑟!]
EX: C(8,4)
8!
𝐶 (8,4) =
=
(8 − 4)! 4!
8!
=
4! 4!
40320
=
24(24)
40320
= 70
576
*There is a combination feature
on your calculator.
EX: Find C(6,3) on your own.
We will use this property of a
combination with the Binomial
Theorem.
EX: Expand (𝑎 + 𝑏)2
We would most likely use FOIL.
(𝑎 + 𝑏)(𝑎 + 𝑏) =
𝑎2 + 𝑎𝑏 + 𝑎𝑏 + 𝑏 2
= 𝑎2 + 2𝑎𝑏 + 𝑏 2
EX: Expand (𝑎 + 𝑏)3
(𝑎 + 𝑏)(𝑎 + 𝑏)(𝑎 + 𝑏)
= (𝑎2 + 2𝑎𝑏 + 𝑏 2 )(𝑎 + 𝑏)
= 𝑎3 + 2𝑎2 𝑏 + 𝑎𝑏 2 + 𝑎2 𝑏 + 2𝑎𝑏 2 + 𝑏 3
= 𝑎3 + 3𝑎2 𝑏 + 3𝑎𝑏 2 + 𝑏 3
We can continue this pattern for
any higher exponent, but how
long would it take to find
(𝑥 + 𝑦)12 ?
There is a quicker way to
perform binomial expansion –
Pascal’s Triangle.
(draw Pascal’s Triangle on the
board)
EX: Find (𝑥 + 𝑦)5 using Pascal’s
Triangle.
1𝑥 5 + 5𝑥 4 𝑦 + 10𝑥 3 𝑦 2 + 10𝑥 2 𝑦 3 + 5𝑥𝑦 4 + 1𝑦 5
Pascal’s Triangle can also be
written using combinations.
(show this on the whiteboard)
EX: Use Pascal’s Triangle to
expand (𝑥 + 2)4
The 4th row of Pascal’s Triangle
is 1,4,6,4,1.
1(𝑥)4 (2)0 + 4(𝑥)3 (2)1 + 6(𝑥)2 (2)2
+ 4(𝑥)1 (2)3 + 1(𝑥)0 (2)4
= 𝑥 4 + 8𝑥 3 + 24𝑥 2 + 32𝑥 + 16
Try the following on your own:
a.) Expand (𝑦 − 2)4
b.) Expand (2𝑥 − 𝑦)4
a.) 𝑦 4 − 8𝑦 3 + 24𝑦 2 − 32𝑦 + 16
b.) 16𝑥 4 − 32𝑥 3 𝑦 + 24𝑥 2 𝑦 2 − 8𝑥𝑦 3 + 𝑦 4
EX: Write the binomial
expansion for (3 − 𝑥 2 )4
1(3)4 (−𝑥 2 )0 + 4(3)3 (−𝑥 2 )1 + 6(3)2 (−𝑥 2 )2 + 4(3)1 (−𝑥 2 )3 + 1(3)0 (−𝑥 2 )4
= 1(81)(1) + 4(27)(−𝑥 2 ) + 6(9)𝑥 4 + 4(3)(−𝑥 6 ) + 1(1)𝑥 8
= 81 − 108𝑥 2 + 54𝑥 4 − 12𝑥 6 + 𝑥 8
For the next few questions, we
will want to use the
combination method for Pascal’s
Triangle.
EX: Find the 5th term of (𝑎 + 2𝑏)8 .
Begin by determining the
variable component.
The 5th term must have variables
𝑎4 𝑏4 .
Now find the coefficient of the
5th term.
𝐶 (8,4)(𝑎)4 (2𝑏)4 = 70𝑎4 (16𝑏 4 )
= 1120𝑎4 𝑏4
EX: Find the third term in the
expansion of (2𝑎 − 3𝑏)11
𝐶 (11,2)(2𝑎)9 (−3𝑏)2
= 55(512𝑎9 )(9𝑏 2 )
= 253440𝑎9 𝑏 2
Expand each binomial on your
own:
a.) (𝑚 + 𝑡)6 b.) (4𝑚 + 2𝑦)4
c.)(2 − 𝑞)5 d.)(𝑥 − 𝑦)3
a.) 𝑚6 + 6𝑚5 𝑡 + 15𝑚4 𝑡 2 +
20𝑚3 𝑡 3 + 15𝑚2 𝑡 4 + 6𝑚𝑡 5 + 𝑡 6
b.) 256𝑚4 + 512𝑚3 𝑦 + 384𝑚2 𝑦 2 +
128𝑚𝑦 3 + 16𝑦 4
c.) 32 − 80𝑞 + 80𝑞2 − 40𝑞3 +
10𝑞4 − 𝑞5
d.) 𝑥 3 − 3𝑥 2 𝑦 + 3𝑥𝑦 2 − 𝑦 3
Find the indicated term of each
expression on your own:
a.)
b.)
a.)
b.)
Fourth term of (2𝑎 + 𝑏)5
Fifth term of (𝑦 − 5)6
40𝑎2 𝑏 3
9375𝑦 2
Let’s write a program to find
each row in Pascal’s Triangle.
Now let’s write a program to
find a specific coefficient in
Pascal’s Triangle.
Upon completion of this lesson,
you should be able to:
1. Evaluate a combination.
2. Expand polynomials using
the Binomial Theorem.
3. Expand polynomials using
Pascal’s triangle.
For more information, visit
http://www.purplemath.com/modules/bino
mial.htm
HW Pg.688 3-60 3rds, 76