MODULAR-TOOL-M-101-PRELIM-PERIOD.docx - College of Teacher...

College of Teacher Education
2 nd Semester A.Y. 2021-2022
MODULAR TOPICS IN M-101 (PRELIM PERIOD)
MODULE 1 . BINOMIAL THEOREM
APPLICABLE FORMULAS
A. PASCAL TRIANGLE 1 (x+y)
0
= 1
1 1 (x+y) 1 = x + y
1 2 1 (x + y) 2 = x 2 + 2xy + y 2
1 3 3 1 (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3
1 4 6 4 1 . . . . . . . . .
1 5 10 10 5 1 (x+y) 5 = x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + y 5
B. BINOMIAL THEOREM FORMULA
(x + y)
n
=
x
n
o!
+
nx
n− 1
y
1
1 !
+
n ( n− 1 ) x
n− 2
y
2
2 !
+
n ( n− 1 )( n− 2 ) x
n− 3
y
3
3 !
+ …… + …… +
y
n
Example. (x + y) 3 =
x
3
0 !
+
3 x
3 − 1
y
1
1 !
+
3.2 x
3 − 2
y
2
2 !
+
y
3
(x +y) 3 =
x
3
1
+
3 x
2
y
1
+
6 x y
2
2
+ y 3
(
x + y
¿¿
3
= x 3 + 3x 2 y + 3xy 2 + y 3
C. COMBINATORIAL EXPANSION FORMULA
( x + y )
n
=
nCo
❑❑
x n--0 + n C
1 x
n
-
1
y 1 + n C
2
x n - 2 y 2 + …………….. + n C n x
n - n
y n
Note that n C r =
n!
r! ( n−r ) !
or you may use your calculator
Example. ( x + y) 5 = 5C
0
x 5 - 0 +
5
C
1
x 5 - 1 y 1 + 5C
2
x 5 - 2 y 2 + 5C
3
x 5 - 3 y 3 + 5C
4
x 5 - 4 y 4 + 5C
5
x 5 - 5 y 5
( x + y)
5
= 1 x
5
+ 5 x
4
y + 10 x
3
y
2
+ 10 x
2
y
3
+ 5 x y
4
+ 1 x
0
y
5
( x+y) 5 = x 5 + 5 x 4 y + 10 x 3 y 2 + 10 x 2 y 3 + 5 x y 4 + y 5
Assignment/Student Activity
Show the complete and detailed solution.
1. Using Pascal Triangle, evaluate and simplify the following
a) ( x + y) 6 b) ( 2x - 3y) 4
2. Using the Binomial Theorem Formula, evaluate and simplify the following
a) ( x -y ) 4 b) ( 3a + 4b) 5
3. Using Combinatorial Expansion Formula, evaluate and simplify the following.
a) (2p + 3q) 4 b) (3m - 5n) 6

ADDITIONAL EXAMPLES
1. Using PASCAL TRIANGLE
Note: The signs are alternately changing from + to - if the binomial form is ( x - y )
n
and the signs are all + if the binomial form is (x + y) n.