Algebraic Expressions Questions

Algebraic Expressions Questions and solutions are provided here to help students of Class 7 to Class 10. As we know, algebraic expressions is one of the most important concepts of mathematics as it deals with the representation of real-life situations mathematically. Also, we can perform various arithmetic operations on algebraic expressions. Let’s understand how to solve various problems related to algebraic expressions and get the practice questions to improve your problem-solving skills.

What is an algebraic expression?

An algebraic expression is formed from variables and constants using different operations. Also, algebraic expressions are made up of terms, and each term is the product of factors. These factors may be numerical or algebraic. Let us consider an example of an algebraic expression.

4x² + 3x – 5

Here,

Terms: 4x², 3x, 5

Arithmetic operations: +, –

Coefficients: 4, 3

Constant: 5

Variable: x

Learn more about algebraic expressions.

Algebraic Expressions Questions

1. Simplify the algebraic expression and write the coefficients:

2x²(x + 2) – 3x (x² – 3) – 5x(x + 5)

Solution:

2x²(x + 2) – 3x (x² – 3) – 5x(x + 5)

= 2x³ + 4x² – 3x³ + 9x – 5x² – 25x

= 2x³ – 3x³ – 5x² + 4x² + 9x – 25x

= -x³ – x² – 16x

Here,

2. Evaluate algebraic expression ax² + by² – cz for x = 1, y = -1, z = 2, a = -2, b = 1, c = -2:

Solution:

Given algebraic expression is:

ax² + by² – cz

Substituting x = 1, y = -1, z = 2, a = -2, b = 1 and c = -2 in the given expression, we get;

ax² + by² – cz = (-2)(1)² + (1)(-1)² – (-2)(2)

= -2 + 1 + 4

= 3

3. Add 3xy + 5yz – 7xz + 1 and -4xy + 2yz – 2xz + 5xyz + 1.

Solution:

(3xy + 5yz – 7xz + 1) + (-4xy + 2yz – 2xz + 5xyz + 1)

Let us group the like terms and then add them.

= (3xy – 4xy) + (5yz + 2yz) + (-7xz – 2xz) + 5xyz + (1 + 1)

= -xy + 7yz – 9xz + 5xyz + 2

4. Write the following statements in terms of algebraic expressions.

(i) Add 4 to the product of a number and 7.

(ii) Subtract 4 from the product of a number and 7.

(iii) Add a number to the product of that number and 6.

(iv) Subtract a number from the product of that number and 8.

(v) Add a number to the product of that number and negative 5.

Solution:

(i) Add 4 to the product of a number and 7 = 7x + 4

(ii) Subtract 4 from the product of a number and 7 = 7x – 4

(iii) Add a number to the product of that number and 6 = 6x + 6

(iv) Subtract a number from the product of that number and 8 = 8x – x

(v) Add a number to the product of that number and negative 5 = -5x + x

5. Simplify the expression: −2a(a + b) − 2a − (a + b)(−2a) − a − 2

Solution:

−2a(a + b) − 2a − (a + b)(−2a) − a − 2

Now, expand the terms.

-2a² – 2ab – 2a – (-2a² – 2ab) – a – 2

= -2a² – 2ab – 2a + 2a² + 2ab – a – 2

Group the like terms and simplify.

= (-2a² + 2a²) + (2ab – 2ab) + (-2a – a) – 2

= 0 + 0 – 3a – 2

= -3a – 2

Therefore, −2a(a + b) − 2a − (a + b)(−2a) − a − 2= -3a – 2.

6. Add the following algebraic expressions.

5x³ + 7 + 6x – 5x², 2x² – 8 – 9x, 4x – 2x² + 3x³, 3x³ – 9x – x² and x – x² – x³ – 4

Solution:

Given,

5x³ + 7 + 6x – 5x², 2x² – 8 – 9x, 4x – 2x² + 3x³, 3x³ – 9x – x² and x – x² – x³ – 4

Now,

(5x³ + 7 + 6x – 5x²) + (2x² – 8 – 9x) + (4x – 2x² + 3x³) + (3x³ – 9x – x²) + (x – x² – x³ – 4)

Combining the like terms, we get;

(5x³ + 3x³ + 3x³ – x³) + (– 5x² + 2x² – 2x² – x² – x²) + (6x – 9x + 4x – 9x + x) + (7 – 8 – 4)

= 10x³ – 7x² – 7x – 5

7. Subtract the sum of – 3x³y² + 2x²y³ and – 3x²y³ – 5y⁴ from x⁴ + x³y² + x²y³ + y⁴.

Solution:

Let us find the sum of – 3x³y² + 2x²y³ and – 3x²y³ – 5y⁴.

i.e., (– 3x³y² + 2x²y³) + (– 3x²y³ – 5y⁴)

= -3x³y² + (2x²y³ – 3x²y³) – 5y⁴

= -3x³y² – x²y³ – 5y⁴

Now, subtract the above sum from x⁴ + x³y² + x²y³ + y⁴.

i.e., (x⁴ + x³y² + x²y³ + y⁴) – (-3x³y² – x²y³ – 5y⁴)

= x⁴ + (x³y² + 3x³y²) + (x²y³ + x²y³)+ (y⁴ + 5y⁴)

= x⁴ + 4x³y³ + 2x²y³ + 6y⁴

8. What should be added to 3pq + 5p²q² + p³ to get p³ + 2p²q² + 4pq?

Solution:

In order to find the required expression, we should subtract 3pq + 5p²q² + p³ from p³ + 2p²q² + 4pq.

Thus, the required expression will be:

p³ + 2p²q² + 4pq − (3pq + 5p²q² + p³)

= p³ + 2p²q² + 4pq − 3pq − 5p²q² − p³

= p³ − p³ + 2p²q² − 5p²q² + 4pq − 3pq

= −3p²q² + pq

Therefore, −3p²q² + pq to be added to 3pq + 5p²q² + p³, to get p³ + 2p²q² + 4pq.

9. Subtract 4.5x⁵ – 3.4x² + 5.7 from 5x⁴ – 3.2x² – 7.3x.

Solution:

Let us subtract 4.5x⁵ – 3.4x² + 5.7 from 5x⁴ – 3.2x² – 7.3x.

(5x⁴ – 3.2x² – 7.3x) – (4.5x⁵ – 3.4x² + 5.7)

= 5x⁴ – 3.2x² – 7.3x – 4.5x⁵ + 3.4x² – 5.7

= 5x⁴ + (3.4x² – 3.2x²) – 7.3x – 4.5x⁵ – 5.7

= 5x⁴ + 0.2x² – 7.3x – 4.5x⁵ – 5.7

= -4.5x⁵ + 5x⁴ + 0.2x² – 7.3x – 5.7

10. Factorise the expression 10x² + 5x + 2xy + y.

Solution:

10x² + 5x + 2xy + y

Take the common factors out.

= 5x(2x + 1) + y(2x + 1)

Again, take the common terms out.

= (2x + 1)(5x + y)

Therefore, 10x² + 5x + 2xy + y = (2x + 1)(5x + y).

Practice Questions on Algebraic Expressions

1. Find the value of the expression a² + 3b² + 6ab for a = 1 and b = – 2.
2. Find the number of terms of the expression 3x²y – 2y²z – z²x + + 4xy – 5.
3. Simplify the expression 50x³ – 21x + 107 + 41x³ – x + 1 – 93 + 71x – 31x³.
4. Add the following expressions:

   t – t² – t³ – 14; 15t³ + 13 + 9t – 8t²; 12t² – 19 – 24t and 4t – 9t² + 19t³

5. How much is y⁴ – 12y² + y + 14 greater than 17y³ + 34y² – 51y + 68?