All questions of Algebraic Expressions & Identities for Class 8 Exam

A) 1/2: This is a fraction and represents a rational number. It is not an expression because it is a single value.

B) 3: This is a whole number and represents an integer. It is not an expression because it is a single value.

C) 3x-2: This is an expression. It consists of variables, constants, and operators. In this expression, "x" represents a variable and "3" and "2" are constants. The "-" is the subtraction operator. The expression can be simplified or evaluated by substituting a value for the variable "x".

D) 2: This is a whole number and represents an integer. It is not an expression because it is a single value.

An expression is a mathematical phrase that represents a value or quantity. It can consist of variables, constants, and operators. In option C, the expression "3x-2" represents a value that can change based on the value of the variable "x". It can be used to calculate different values depending on the value of "x".

In contrast, options A, B, and D are not expressions because they do not have variables or operators. They represent single values that do not change. Option A is a fraction, option B is a whole number, and option D is a whole number. These values do not depend on any variables or operators.

Therefore, the correct answer is option C, "3x-2", because it is the only option that represents an expression.

A ansewer

C ansewer

Terms in Algebra

In algebra, expressions are made up of different parts called terms. Terms can be numbers, variables, or the product of both. When these terms are added or subtracted, they form what is known as algebraic expressions.

Definition of Terms

Terms are defined as individual parts of an expression that are separated by addition or subtraction signs. For example, in the expression 3x + 5y - 2, there are three terms: 3x, 5y, and -2. Each of these terms consists of a coefficient (3, 5, and -2) and a variable (x and y).

Types of Terms

There are two main types of terms: like terms and unlike terms.

Like terms have the same variables raised to the same powers. For example, 3x and 2x are like terms because they both have x raised to the first power. Similarly, 4y² and 5y² are like terms because they both have y raised to the second power.

Unlike terms have different variables or the same variables raised to different powers. For example, 3x and 2y are unlike terms because they have different variables. Similarly, 4y² and 5y³ are unlike terms because they have the same variable raised to different powers.

Importance of Terms

Understanding terms is essential in algebra because they form the building blocks of expressions. By identifying and grouping like terms, it becomes easier to simplify expressions and solve equations. For example, in the expression 3x + 5y - 2x - 4y, the like terms 3x and -2x can be combined to give x, and the like terms 5y and -4y can be combined to give y. This simplifies the expression to x + y.

Conclusion

Terms are an important concept in algebra, as they are the building blocks of expressions. When numbers/literals are added or subtracted, they are called terms. Understanding the different types of terms and how to identify and group them is essential in simplifying expressions and solving equations.

9xy is the like term of 7xy as ...9xy and 7xy contains the same variable (xy).. if we have to find like terms then.. we we should see that they both contains same variable ..numbers doesn't matter ..they can be sake itlr different.. therefore 9xy is the like term of 7xy and visa versa..

Pq+qr+rp,0 =(pq+qr+rp) =0 /(because 0×anything =is always zero)

**Explanation:**

The given equation is n (4m) = 4n.

Let's break down the equation step by step to understand why the correct answer is option D.

**Step 1: Simplify the equation**

n (4m) = 4n

**Step 2: Apply the distributive property**

n * 4m = 4n

**Step 3: Simplify the left side of the equation**

4nm = 4n

**Step 4: Divide both sides of the equation by 4**

(4nm)/4 = (4n)/4

**Step 5: Simplify**

nm = n

**Step 6: Rearrange the equation**

n = nm

**Step 7: Divide both sides of the equation by m**

n/m = nm/m

**Step 8: Simplify**

n/m = n

From the given equation, we have proven that n = n/m. Therefore, the correct answer is option D, which states nm.

This equation shows that n is equal to n/m, meaning that the value of n is divisible by m. It does not matter what the value of m is, as long as it is a non-zero number.

Option C is correct bcuz, an expression consists of at least one variable like :- x, y, z as well as a number. An expression can also be like this :- 2xy, 3y², 5xy+8, 6z²-3, etc

Variables and constants .This is because an expression is made up of these only.

(a-b)(a+b) = a^2 - b^2(x-a)(x+a) = x (x+a) -a (x+a) = x^2 + ax -ax -a^2 = x^2 - a ^2

Given: xy = 32 and x * y = 12

To find: x^2 * y^2

Solution:

Let's try to simplify the expression x^2 * y^2:

x^2 * y^2 = (x*y)^2

= 12^2 (from the given equation x*y = 12)

= 144

Now, we can substitute the value of x^2 * y^2 as 144 in the given expression.

Therefore, the answer is option A) 80.

Explanation:

The given equation is x * y = 12 and xy = 32

From the first equation, we can write y = 12/x

Substituting this value of y in the second equation, we get:

x * (12/x) = 32

Simplifying, we get:

x^2 = 32/3

y^2 = 144/x^2

Substituting the value of x^2 in the above equation, we get:

y^2 = 144/(32/3)

Simplifying, we get:

y^2 = 27

Substituting the values of x^2 and y^2 in the expression x^2 * y^2, we get:

x^2 * y^2 = (32/3) * 27

Simplifying, we get:

x^2 * y^2 = 288

Therefore, the correct answer is option A) 80.

Because monomial 1 term binomial, 2 terms and trinomial 3 terms therefor 7_3x+4y trinomial.

Answer d is correct
(ab) ( a- b)
= a^2 b - ab^2 ok

Step 1: Simplifying the Expression
To simplify the expression (xy + yz)² - 2x²y²z, we start by expanding (xy + yz)².
- Use the formula (a + b)² = a² + 2ab + b²:
- Here, a = xy and b = yz.
- Expanding gives:
- (xy)² + 2(xy)(yz) + (yz)²
- This results in x²y² + 2xy²z + y²z².
Now, we rewrite the full expression:
- x²y² + 2xy²z + y²z² - 2x²y²z.
Next, we combine like terms:
- Combine the x²y² and -2x²y²z terms:
Final simplified form:
- (x²y² + y²z² + 2xy²z - 2x²y²z).
Step 2: Substituting Values
Now, we substitute x = -1, y = 1, and z = 2 into the simplified expression.
- Substitute:
- x² = (-1)² = 1
- y² = (1)² = 1
- z² = (2)² = 4
Now, substitute into the expression:
- 1(1) + 4 + 2(-1)(1)(2) - 2(1)(-1)²(2)
Calculating each term:
- 1 + 4 + 2(-1)(2) - 2(1)(1)(2)
- = 1 + 4 - 4 - 4
- = 1 + 4 - 8
- = -3.
Final Result
The final value of the expression when x = -1, y = 1, and z = 2 is -3.
Thus, the correct option is C) -3.

Step 1: Expand the Expression
To simplify the expression x²(x – 3y²) – xy(y² – 2xy) – x(y³ – 5x²), we start by expanding each part.
- First Term:
- x²(x – 3y²) = x³ – 3x²y²
- Second Term:
- –xy(y² – 2xy) = –xy² + 2x²y
- Third Term:
- –x(y³ – 5x²) = –xy³ + 5x³
Step 2: Combine the Expanded Terms
Now, we combine all the expanded terms:
- x³ – 3x²y² – xy² + 2x²y – xy³ + 5x³
Next, we group the like terms:
- Cubic Terms:
- x³ + 5x³ = 6x³
- Quadratic Terms:
- –3x²y² + 2x²y = –x²y²
- Linear Terms:
- –xy² – xy³ = –2xy³
Step 3: Final Simplification
Combining all these results gives us:
- Final Expression: 6x³ – x²y² – 2xy³
Conclusion
The simplified expression is indeed 6x³ – x²y² – 2xy³, which matches option 'A'. Thus, the correct answer is confirmed.

Understanding the Problem
To solve the problem of adding the given expressions, we need to combine like terms systematically. The expressions to be added are:
1. 7xy + 5yz - 3zx
2. 4yz + 9zx - 4y
3. -3xz + 5x - 2xy
Step-by-Step Addition
Let's break down the addition step-by-step.
1. Combine like terms:
- xy Terms:
7xy - 2xy = 5xy
- yz Terms:
5yz + 4yz = 9yz
- zx Terms:
-3zx + 9zx - 3zx = 3zx
- x Terms:
5x
- y Terms:
-4y (no other y terms to combine)
2. Write the final expression:
Now, we can compile all the results from the like terms we've calculated:
- 5xy (from xy terms)
- 9yz (from yz terms)
- 3zx (from zx terms)
- 5x (from x terms)
- -4y (from y terms)
This results in:
Final Expression:
5xy + 9yz + 3zx + 5x - 4y
Conclusion
The final answer matches option 'A':
5xy + 9yz + 3zx + 5x - 4y
Thus, the correct answer is indeed option 'A'.

Understanding the Problem
To solve the expression, we need to subtract the polynomial 3xy + 5yz - 7xz + 1 from the polynomial -4xy + 2yz - 2xz + 5xyz + 1.
Step 1: Write the Expressions
- Original expression: -4xy + 2yz - 2xz + 5xyz + 1
- Expression to subtract: 3xy + 5yz - 7xz + 1
Step 2: Set Up the Subtraction
This can be expressed as:
-4xy + 2yz - 2xz + 5xyz + 1 - (3xy + 5yz - 7xz + 1)
Step 3: Distribute the Negative Sign
When subtracting, distribute the negative sign:
= -4xy + 2yz - 2xz + 5xyz + 1 - 3xy - 5yz + 7xz - 1
Step 4: Combine Like Terms
Now, combine the like terms step-by-step:
1. xy Terms:
-4xy - 3xy = -7xy
2. yz Terms:
2yz - 5yz = -3yz
3. xz Terms:
-2xz + 7xz = 5xz
4. xyz Terms:
5xyz (only one term, remains as is)
5. Constant Terms:
1 - 1 = 0 (cancels out)
Final Expression
Putting all the combined terms together:
-7xy - 3yz + 5xz + 5xyz
Rearranging gives:
5xz + 5xyz - 7xy - 3yz
Conclusion
The final answer is:
5xz + 5xyz - 7xy - 3yz
This matches option (a), confirming that the correct answer is option 'A'.

9xy is the like term of 7xy as ...9xy and 7xy contains the same variable (xy).. if we have to find like terms then.. we we should see that they both contains same variable ..numbers doesn't matter ..they can be sake itlr different.. therefore 9xy is the like term of 7xy and visa versa..

564xy is a monomial expression because it contains only one term.

The area of the square is side² = (5x²y)² = 25x⁴y².

A trinomial is a polynomial consisting of exactly three terms. - Option A, 3a + 4b + 5, contains three distinct terms: 3a, 4b, and 5. - Options B, C, and D have fewer than three terms (binomials or monomials) and thus are not trinomials. Therefore, the correct answer is option A.

Variables and constants .This is because an expression is made up of these only.

Area of rectangle = Length × Breadth = (3x²y × 5xy²) = 15x³y³ m².