Simplify the expression [1/(3x + 3h) – (1/3x)]/h

Algebraic Expression is made up of variables and constants along with algebraic operations such as addition, subtraction, etc.. these Expressions are made up of terms. Algebraic expressions are the equations when the operations such as addition, subtraction, multiplication, division, etc. are operated upon any variable.

The above expressions are represented with the help of unknown variables, constants, and coefficients. The combination of these three terms is termed as an expression. Unlike the algebraic equation, it has no sides or ‘equals to’ sign.

Types of Algebraic Expression

A combination of terms by the operations such as addition, subtraction, multiplication, division, etc is termed as an algebraic expression (or) a variable expression. Examples: 2x + 4y – 7, 3x – 10, etc. Algebraic Expressions are of three types based on the number of terms in the expression. 

• Monomial Expression: An expression that has only one term is termed a Monomial expression.

Examples of monomial expressions include 5x⁴, 3xy, 2x, 5y, etc.

• Binomial Expression: An algebraic expression which is having two terms and unlike are termed as a binomial expression

Examples of binomial include 2xy + 8, xyz + x², etc.

• Polynomial Expression: An expression that has more than one term with non-negative integral exponents of a variable is termed a polynomial expression.

Examples of polynomial expression include ax + by + ca, x³ + 5x + 3, etc.

Other Types of Expression

Other expressions are also apart from monomial, binomial, and polynomial types of expressions which are,

• Numeric Expression: An expression that consists of only numbers and operations, but never includes any variable is termed a numeric expression.

Some of the examples of numeric expressions are 14 + 5, 18 ÷ 2, etc.

• Variable Expression: An expression that contains variables along with numbers and operations to define an expression is termed a variable expression.

Some examples of a variable expression include 4x + y, 5ab + 53, etc.

Some Important Algebraic Formulas

There are some terms of algebraic expression which basically used,

1. (a + b)² = a² + 2ab + b²
2. (a – b)² = a² – 2ab + b²
3. (a + b)(a – b) = a² – b²
4. (x + a)(x + b) = x² + x(a + b) + ab
5. (a + b)³ = a³ + b³ + 3ab(a + b)
6. (a – b)³ = a³ – b³ – 3ab(a – b)
7. a³ – b³ = (a – b)(a² + ab + b²)
8. a³ + b³ = (a + b)(a² – ab + b²)

Simplify the expression [1/(3x + 3h) – (1/3x)]/h

Solution: 

Given term:  {{1}/{3x + 3h} – {1}/{3x}}/h}

By simplifying, write 

= {(3x – 3x – 3h) / (3x+3h)(3x)} × (1/h)

= {(-3h) / (9x² + 9xh )} × (1/h)

= {(-3h) / (9x²h + 9xh²)

= {(-3h)} / {9xh (x + h)}

= {-1 /3(x + h)}

Similar Problems

Question 1: If 2x² + 3xy + 4x + 7 is an algebraic expression. Determine the equation.

Solution:

2x², 3xy, 4x, and 7 are the Terms.

Coefficient of term: 2 is the coefficient of x²

Constant term: 7

Variables: here x, y are variables

Factors of a term: If 2xy is a term, then its factors are 2, x, and y.

Like and Unlike terms: Example of like and unlike terms:

• Like terms: 4x and 3x
• Unlike terms: 2x and 4y

Question 2: Simplify: 7 – 2(x – 1).

Solution:

Here we have

7 – 2(x – 1)

= 7 – 2x + 2

= 9 – 2x

= -2x + 9

Question 3: Simplify 5x² + 7x – 9 = 4x² + x – 18

Solution:

5x² + 7x -9 = 4x² + x – 18

5x² + 7x -9 – 4x² – x + 18 = 0

x² +6x + 9 = 0                                          

{(a + b)² = a² + 2ab + b²}

(x + 3)² = 0                     

Question 4:  Divide and simplify, (21x³ – 7)/(3x – 1)

Solution:

(21x³ – 7)/(3x – 1)

= [7 (3x³ – 1)] / (3x – 1)

= [7 {(3x)³ – (1)³ ] / (3x-1)

= [7 (3x – 1)(9x² + 1 + 3x)] / (3x – 1)                   

{a³ – b³ = (a – b)(a² + ab + b²)}

= 7 (9x² +1 + 3x)

= 63x² + 7 + 21x

Question 5: Factorize 6a(a + 6)^2/3 + 8(a + 6)^1/3

Solution:

Given: [6a(a + 6)^2/3] + [8(a + 6)^1/3]

From above expression we will factorize,

= [2.3a(a + 6)^2/3] + [(2)³ (a + 6)^1/3]

= 2(a + 6)^1/3 [{3a(a + 6)^1/3 + 2²]

=  2(a + 6)^1/3 {3a(a + 6)^1/3 + 4}

=  2(a + 6)^1/3 {3a(a + 6)^1/3 + 4}

Question 6: Simplify the expression. {38x²yz²}/{-19xy²z³}?

Solution:

= {38x²yz²}/{-19xy²z³}

Divide like terms

= -(38 / 19) × (x² / x ) × (y / y² ) × ( z² / z³ )

By simplifying

= – 2x / yz

So the final result is – 2x / yz

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