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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