Dr. Fe L. FazDr. Rey S. GuevarraDr. Meredith P. RomeroDr. Emelita D. BautistaMathematics Teachers

Pedro E. Diaz High School

CHECKING OF ASSIGNMENT

How many terms are there in the expansion of (x+y)0

What is the second term in the expansion of (a+i)3

Find the indicated term in the expansion of each given expression: 4th term; (x + y)5

2th term; (p + q)6

1st term; (x + y)2

Direction: Answer the following

Problem of the day!!!Monkey Donkey Paradox

On the first day, monkey donkey ate 1 piece of cupcake. On the 2nd day, monkey donkey ate 1 cupcake at the morning and 1 more during nighttime for a total of 2 cup cakes. On the third day, monkey donkey ate 1 cup cake at the morning, 2 at lunch time and 1 more during night time for a total of 4 cup cakes. On the fourth day, monkey donkey ate 1 cup cake, then 3 cup cakes and 3 more, then 1 more at the end of the day, for the total of 8 cup cakes. If this pattern continues, how many cup cakes will monkey donkey eat on the 5th day? On the 6th day?

PASCAL’S TRIANGLE

A French mathematician, who discovered a pattern row known as Pascal’s Triangle of Coefficients.

BLAISE PASCAL(1623 - 1662)

used to find the coefficients of the expansion of any integral power.

PASCAL TRIANGLE

(x + y)0

(x + y)1

(x + y)2

(x + y)3

(x + y)4

(x + y)5

(x + y)6

(x + y)7

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

The coefficients may be written in either way

(x + y)0

(x + y)1

(x + y)2

(x + y)3

1

1 1

1 2 1

1 3 3 1

1 9 36 84 126 126 8436 9 1

0

1

2

3

4

5

6

7

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

8 1 8 28 56 70 56 28 8 1

9

1 10 45 120 210 252 210 120 45 10 1

10

Illustrative Examples: 1.What is the fourth term

when (x + y)7 is expanded?

Solution:

8th row: 1, 7, 21, 35, 35, 21, 7, 1

x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7

4th term: 35x4y3

2. What is the third term in the expansion of (a + i)5 ?

Solution:

6th row: 1, 5, 10, 10, 5, 1

a5 + 5a4i + 10a3i2 + 10a4i3

+ 5a3i4 + i53rd term: 10a3i2

3. What is the sum of the numerical coefficients when (x + y)6 is expanded?

Solution:

7th row: 1, 6, 15, 20, 15, 6, 1

1 + 6 + 15 + 20 + 15 + 6 + 1 = 64

The sum of the coefficients: 64

Group Activity

Activity: Direction: Write the expanded form of each

binomial expression and identify the term asked:

1.(x + y)4; second

term

2. (x + y)8; fourth term

3. (x + y)10; sixth term

4. (x + y)6; third term

Summary

What are the characteristics of the

product of the binomial expression (x + y)n ,

where n represents the integral exponent?

Let’s try some challenge…

Choose the letter of the best answer.

The essence of Mathematics is not

to make things complicated but to make complicated

things simple.

AgreementThink of this: Add the terms in each of

the first five rows of the Pascal’s Triangle. Compare the sum and find a pattern for this sequence. Make a general formula to express this relation.