square roots of algebraic expression

Square roots of Algebraic Expression

Square roots of Algebraic Expression

- Identify the algebraic equation

Introduction

Homogenous Expression

Definition and Example of Homogenous Expression

Symmetric Expression

Definition of Symmetric Expression

Homogeneous Symmetric Expression

Definition and Example of Homogeneous Symmetric Expression

Cyclic Expression

Definition and Example of Homogeneous Symmetric Expression

Summary

Assessment

Introduction

Animation Description

School playground.

VO

Let us consider a situation- In a school playground students are standing in rows.


Students are standing in equal rows.

VO

Each row has equal number of students and this number is equal to the total number of rows.


Number(4) will appear on the screen.Text will appear one by one.

VO

If the number of rows to be formed is 4, and each row have 4 students, then the total no. of students in the school is 4X4 = 16 which is equal to 42 square root of 16 is 4 i.e.√16=4Where ‘ √’ is the sign used for representing square root.

4

4

If the number of rows to be formed is 4,

and each row have 4 students, then the total no. of students in the school is 4X4 = 16

which is equal to 42 square root of 16 is 4 i.e.√16=4

Where ‘ √’ is the sign used for representing square root.

Homogenous Expression


Students of 3 different schools will appear through animated manner.

VOStudents of 3 different schools standing in a playground. The no. of students of the 3 different schools may not be equal but the standing arrangement of each school should be in such way that the no. of students in a row should be equal to the no. of rows.This arrangement is called homogeneous Expression.


The screen will be represented through text animation.

Definition of Homogenous Expression :–

If we consider the previous situation, then each row of students(in algebraic expression is a term) is a term of an algebraic expression and the number of students in those rows is the degree of that row (term). Such an expression is called homogeneous expression. Therefore, if all the terms of an algebraic expression are of some degree , then such an expression is called a homogeneous expression.

Example : ax2 +2hxy+by2

Example :1)Is x2+xy+y2 is a homogeneous expression?

Term Degree1st term is x2 whose degree is 2 x2 2 2nd term is xy whose degree 1+1=2 xy 2 3rd term is y2 whose degree is 2 y2 2

So, it is a homogeneous expression

SAME


The screen will be represented through text animation. Example :

2) Is ax+by+c is a homogeneous expression?

Term Degree

1st term is ax whose degree is 1 ax 1 2nd term is by whose degree 1 by 1 3rd term is c whose degree 0 c 0

So, it is not a homogeneous expression.

NOT SAME

Symmetric Expression


Students will appear by animated manner.

VOBoys and girls of a school are made to stand in separate rows: Now some boys are asked to join the girls’ rows and some girls are asked to join the boys’ rows, keeping the no. of students in each row same. If the previous arrangement matches the new one, then the 1st situation represents symmetric expression.



Definition of Symmetric Expression :–

Taking the previous situation into consideration: if boys represent x & girls represent y then the first arrangement can be termed as f (x,y) . Similarly, the second arrangement can be termed as f(y,x). If f(x,y) = (match f(y,x), then f(x,y) is a symmetric expression eg: f(x,y) = ax+b+ay f(y,x) = ay+b+ax [ putting y in place of x & vice- versa] So, f(x,y) is symmetric.

Examples

1)Is x3+3x2y+xy2+y3 a symmetric expression? We assume x3+3x2y+xy2+y3 as f(x,y).By putting y in place of x & vice-versa, we get f(y,x)=y3+3y2x+yx2+x3

Thus, f(x,y) ≠ f(y,x) and f(x,y) is not a symmetric expression. 2)Is f(x,y) = x3+y3+x+y is a symmetric expression?Putting y in place of x and vice-versa in f(x,y), we get f(y,x) = y3+x3+y+xAs f(x,y)=f(y,x), f(x,y) is a symmetric expression.

Homogeneous Symmetric Expression


Students will appear by animated manner.

VOLet us consider a situation: Students of two classes are made to stand in lines according to the class in which they study. The no. of students in each line should be equal to the no.of lines.o t

Condition of homogeneous expression is fulfilled. the no. of lines..


Students will appear in animated manner

VOStudents of one class interchange their positions with the students of another class. If the two arrangements are same. Condition of symmetric expression is fulfilled.This whole scenario is called homogeneous symmetric expression.




An expression which fulfills the conditions of both homogeneous and symmetric expressions, is said to be a homogeneous symmetric expression.

Examples

Is f (x,y,z) = ax+ay+az a homogeneous symmetric expression?Degree of each term in f (x,y,z) are equal to 1. So it is a homogeneous expression now by replacing x with y, y with z and z with x, we getf(y,x,z) = ay+az+ax = f(x,y,z) so it is symmetric as well.Hence, f(x,y,z) is a homogeneous symmetric expression.

Cyclic Expression


The screen will be represented through object animation.

VOLet us imagine a wheel with 3 points a,b,c on its circumference and portion of the wheel indicated by b is touched to the ground. If we rotate the wheel along the direction from b to c, then again rotate it along c to a & then from a to b.


The screen will be represented through object & text animation.

VO

Then the/This situation can be represented in the form of an expression as bc+ca+ab

bc+ca+ab


The screen will be represented through object animation.

VO

Similarly if the wheel is rotated from point c, the expression will be ca+ab+bc.As both the situation provide the same expression, such an expression is called cyclic expression where a,b,c,follow a cyclic order.




An expression f(x,y,z) is said to be a cyclic expression if f (x,y,z) = f(y,z,x).To write cyclic expression ∑ (read as sigma) used for sum of terms and π (read pi) used for product of terms. Example :∑ x(x+y) is the short form of the sum of cyclic expressions of the type like y(z+x),z(x+y) & x(y+z).i.e. ∑x(y+z)=x(y+z)+y(z+x)+z(x+y) x,y,z.



Example :1.Simply the cyclic expression: ∑(b-c)(b+c). This is the sum of cyclic expressions of the type (b-c) (b+c), (c-a) (c+a) & (a-b)

(a+b) (a,b) (c,a)

(b,c) Moving in cyclic order

Thus, ∑(b-c) (b+c) = (b-c) (b+c) +(c-a) (c+a) + (a-b) (a+b) = b2-c2+c2-a2+a2-b2 = 0 [after cancelling it comes to 0]

a,b,c We know that (b-c) (b+c) = b2-c2



Example :2. Simplyfy : ( ∑a)2 - ∑a2

a,b a,b

Let us take :( ∑a)2 and ∑a2 as two different functions f1, (a,b) & f2(a,b) respectively a,b a,b

f1, (a,b) = (a+b)

f2, (a,b) = a2+b2

Now, putting these values in the original expression, we get

( ∑a)2 - ∑a2 = (a+b)2 – (a2+b2) a,b a,b

= a2 +2ab+b2-a2-b2 [expanding (a+b)2, we get a2+2ab+b2]

= 2ab


Bulleted point will be appear one by one.

Summary

Considering an algebraic expression f(x,y),If all Terms are of same degree, it is known as homogeneous expression.If f(x,y) = f(y,x), it is known as symmetric expression.If bith the conditions are fulfilled, it is known as homogeneous symmetic expression.

Consider an algebraic expression f(x,y,z) in which x,y,z are replaced in a particular Cyclic order and f(x,y,z) = f(y,z,x), then f(x,y,z) is a cyclic expression.


Click on the correct answer.

Assessment : 1 Choose the correct answer :–

x+y+y2 is a homogeneous expression

YES NO Done

[Answer : NO]

Done No Yes


Click on the correct answer.

Assessment : 1

Choose the correct answer :–

x2+y2+x+y+1 is a

[Answer : Symmetric expression]

Done

Homogeneous expression

Symmetric expression

Homogeneous symmetric expression

Cyclic expression


Check box activity.

Assessment : 1 Drag and drop :–

Correct cyclic expression for π (z+c) is a,b,c

( ) X ( ) X ( )

z+a a+b z+b z+c c+z z+c

Answer : (z+a) (z+b)(z+c)

Check Solution

square roots of algebraic expression

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