# GENERATING EQUIVALENT ALGEBRAIC EXPRESSIONS

## About "Generating equivalent algebraic expressions"

Generating equivalent algebraic expressions :

An algebraic expression is a mathematical sentence involving constants (any real number), variables and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).

Let us consider the algebraic expression which is given below.

7(x - 3) + 2(2x - 5) - 3(x - 5)  --------- (1)

To generate equivalent algebraic expression, we have to simplify.

Let us simplify the expression as given below.

Step 1 :

Distributing the number which is outside the parenthesis with inner terms.

=  7x - 21 + 4x - 10 - 3x + 15

Step 2 :

Combine the like terms

=  7x + 4x - 3x - 21 - 10 + 15

= 11x - 3x - 31 + 15

= 8x - 16

Hence, equivalent algebraic expression to (1) is (8x - 16)

## Simplifying rational expressions - Examples

Example 1 :

Write the equivalent algebraic expression for the function defined by

f(x)  =  (x² - 5x + 6) / (x - 3)

Solution :

Example 2 :

Write the equivalent algebraic expression for the expression given below.

4x - (2 + 4x) - 2 (x - 1) - 8 (x -3)

Solution:

=  4x - (2 + 4x) - 2 (x - 1) - 8 (x -3)

=  4x - 2 - 4x - 2x + 2 - 8x + 24

=  4x - 4x - 2x - 8x - 2 + 2 + 24

=  -10x + 24

Example 3 :

Write the equivalent algebraic expression for the function defined by

f(x)  =   (5x + 20) / (7x + 28)

Solution:

Factoring the numerator and denominator of the given rational function, we get

f(x)  =   5(x + 4) / 7(x + 4)

Getting rid of the common factor (x + 4) at both numerator and denominator, we get

f(x)  =  5 / 7

Example 4 :

Write the equivalent algebraic expression for the expression given below.

(2x - 7)/5 + (x + 9)/15 - (2x -2)/5

Solution :

=  (2 x - 7)/5 + (x + 9 )/15 - (2 x - 2 )/5

Now we are going to combine the like terms

Example 5 :

Write the equivalent algebraic expression for the expression given below.

(3x+41)/2 + (x-3 )/5 -(9-2x)/6

Solution :

Now we are going to distribute the numbers which is out side the  parenthesis to the inner terms.Then combine the like terms

Example 6 :

Write the equivalent algebraic expression for the function defined by

f(x)  =   (6x² - 54) / (x² + 7x + 12)

Solution:

Factoring the numerator and denominator of the given rational function, we get

f(x)  =   6(x² - 9) / (x + 3)(x + 4)

f(x)  =   6(x² - 3²) / (x + 3)(x + 4)

f(x)  =   6(x + 3)(x - 3) / (x + 3)(x + 4)

Getting rid of the common factor (x + 3) at both numerator and denominator, we get

f(x)  =  6(x-3) / (x + 4)

Example 7 :

Write the equivalent algebraic expression for the function defined by

f(x)  =   (64a³ + 125b³) / (4a²b + 5ab²

Solution:

f(x)  =   (64a³ + 125b³) / (4a²b + 5ab²

f(x)  =   (4³a³ + 5³b³) / (4a²b + 5ab²)

f(x)  =   [(4a)³ + (5b)³] / (4a²b + 5ab²)

Factoring the numerator and denominator of the given rational function, we get

f(x)  =   [4a + 5b][(4a)² - 4a.5b + (5b)²] / ab(4a + 5b)

f(x)  =   [(4a)² - 4a.5b + (5b)²] / ab

f(x)  =   [16a² - 20ab + 25b²] / ab

Getting rid of the common factor (4a + 5b) at both numerator and denominator, we get

f(x)  =   [(4a)² - 4a.5b + (5b)²] / ab

f(x)  =   [16a² - 20ab + 25b²] / ab

Example 8 :

Write the equivalent algebraic expression for the function defined by

f(x)  =   (x² + 7x + 10) / (x² -  4

Solution:

f(x)  =   (x² + 7x + 10) / (x² -  4)

f(x)  =   (x² + 7x + 10) / (x² - 2²)

Factoring the numerator and denominator of the given rational function, we get

f(x)  =   (x + 2)(x + 5) / (x + 2)(x - 2)

Getting rid of the common factor (x + 5) at both numerator and denominator, we get

f(x)  =   (x + 5) / (x - 2)

After having gone through the stuff given above, we hope that the students would have understood "Generating equivalent algebraic expressions".

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