Algebraic Expressions and Identities Chapter Notes - (Maths) Class 8 PDF
What are Algebraic Expressions?
Any mathematical expression that consists of numbers, variables and operations is called an algebraic expression. Examples: \(3x + 4\), \(5ab - 2b + 7\).
Components of an Algebraic Expression
- Constants: Fixed numerical values. Example: in \(3x + 4\), 4 is a constant.
- Variables: Symbols that represent unknown or varying quantities. Example: \(x\) in \(3x + 4\) is a variable.
- Coefficients: Numbers multiplied by variables. Example: in \(3x\), 3 is the coefficient of \(x\).
- Operators: Symbols that indicate operations: \(+\), \(-\), \(\times\), \(\div\).
Types of Algebraic Expressions
Polynomials are algebraic expressions made up of one or more terms. Terms separated by \(+\) or \(-\) are called unlike terms if their variable parts differ. Common types:
- Monomial: An algebraic expression having only one term, e.g., \(7x^2\), \(-3ab\).
- Binomial: An algebraic expression having two terms, e.g., \(x + 5\), \(3a - 2b\).
- Trinomial: An algebraic expression having three terms., \(x^2 + 2x + 1\).
- Polynomial: Expression with one or more terms where each term is a product of a constant and variables raised to whole-number powers.
Addition and Subtraction of Algebraic Expressions
Method
- Arrange the expressions so that like terms appear in the same column.
- Add or subtract the coefficients of like terms as you would combine like numerical terms.
- If a term is not present in one expression, treat its coefficient as zero and bring the other term unchanged into the result.
Example 1: Add \(15p^{2} - 4p + 5\) and \(9p - 11\).
Solution: Write the expressions one below the other aligning like terms.
Combine like terms.
Example 2: Subtract \(5a^{2} - 4b^{2} + 6b - 3\) from \(7a^{2} - 4ab + 8b^{2} + 5a - 3b\).
Solution: Change the signs of all terms of the expression to be subtracted, then add.
Now add to the first expression.
Combine like terms.
Multiplication of Algebraic Expressions
When multiplying algebraic terms watch how coefficients combine and how exponents add for the same base. Use the distributive law to multiply monomials with polynomials and polynomials with polynomials.
Multiplication of Like Terms
The coefficients multiply and powers of the same variable add.
Example 3: Product of \(4x\) and \(3x\).
Solution: Multiply the coefficients. \(4 \times 3 = 12\)
Example 4: Product of \(5x\), \(3x\) and \(4x\).
Solution: Multiply coefficients: \(5 \times 3 \times 4 = 60\)
Multiplication of Unlike Terms
- Multiply the coefficients.
- If the variables are different, each variable is included unchanged in the product.
- If some variables are the same, add the exponents of those variables.
Example 5: Product of \(2p\) and \(3q\).
Solution: Multiply coefficients: \(2 \times 3 = 6\)
Example 6: Product of \(2x^{2}y\), \(3x\) and \(9\).
Solution: Multiply coefficients: \(2 \times 3 \times 9 = 54\)
Multiplying a Monomial by a Monomial
Multiplying Two Monomials
General rule:
- Coefficient of product = coefficient of first monomial × coefficient of second monomial.
- Variable part = product of variable parts; add exponents for like bases.
Example 7: \( (3x^{2}) \times (4x^{3}) \)
Solution: Multiply coefficients: \(3 \times 4 = 12\)
Example 8: \( (2a^{3}b) \times (5ab^{2}) \)
Solution: Multiply coefficients: \(2 \times 5 = 10\)
Multiplying Three or More Monomials
The same rules extend: multiply all coefficients and add exponents for each variable across all monomials.
Example 9: \(4xy \times 5x^{2}y^{2} \times 6x^{3}y^{3}\)
Solution: Multiply coefficients: \(4 \times 5 \times 6 = 120\)
Example 10: Find the volume of each rectangular box with given length, breadth, and height.
| Length | Breadth | Height |
|---|---|---|
| \(2ax\) | \(3by\) | \(5cz\) |
| \(m^{2}n\) | \(n^{2}p\) | \(p^{2}m\) |
| \(2q\) | \(4q^{2}\) | \(8q^{3}\) |
Volume = length × breadth × height
Solution (i): Multiply coefficients: \(2 \times 3 \times 5 = 30\)
Solution (ii): Multiply like bases: \(m^{2}n \times n^{2}p \times p^{2}m\)
Solution (iii): Multiply coefficients: \(2 \times 4 \times 8 = 64\)
Multiplying a Monomial by a Polynomial
Multiplying a Monomial by a Binomial
Multiply the monomial with each term of the binomial and then add the results (distributive law).
Example 11: Simplify \(2x \times (3x + 5xy)\).
Solution: Multiply \(2x\) by the first term: \(2x \times 3x\)
Example 12: Simplify \(a^{2} \times (2ab - 5c)\).
Solution: Multiply \(a^{2}\) by \(2ab\): \(a^{2} \times 2ab\)
\(= 2a^{3}b\)
Multiplying a Monomial by a Trinomial
Multiply the monomial with each term of the trinomial and add the products.
Example: \(4q \times (5q^{2} + 6q + 8)\)
Solution: \(4q \times 5q^{2} = 20q^{3}\)
Example 13: Simplify and evaluate as directed:
(i) \(x(x - 4) + 3\) for \(x = 2\)
(ii) \(5z(3z - 9) - 4(z - 2) - 45\) for \(z = -1\)
Solution:
(i) Expand: \(x(x - 4) + 3\)
(ii) Expand: \(5z(3z - 9) - 4(z - 2) - 45\)
Try yourself: What is the result of multiplying 5a and 2a2?
Multiplying a Polynomial by a Polynomial
Multiplying a Binomial by a Binomial
Use the distributive law (each term of the first multiplies every term of the second). Combine like terms at the end.
Example 14: Simplify \((3a + 4b) \times (2a + 3b)\).
Solution: Apply distributive law: \(3a \times (2a + 3b) + 4b \times (2a + 3b)\)
Multiplying a Binomial by a Trinomial
Multiply each term of the binomial by each term of the trinomial, then combine like terms.
Example 15: Simplify \((p + q) (2p - 3q + r) - (2p - 3q) r\).
Solution: First expand \((p + q)(2p - 3q + r)\):
What is an Identity?
An identity is an equality that holds true for every permissible value of the variables. An equation is true only for certain values (solutions) and so is not an identity unless it holds for all values of the variable(s).
Example: \(x^{2} = 1\) is an equation true only for \(x = 1\) and \(x = -1\); it is not an identity.
Standard Identities
These identities are widely used to expand squares and products quickly and to simplify algebraic calculations.
- \((a + b)^{2} = a^{2} + 2ab + b^{2}\)
- \((a - b)^{2} = a^{2} - 2ab + b^{2}\)
- \(a^{2} - b^{2} = (a + b)(a - b)\)
- \((x + a)(x + b) = x^{2} + (a + b)x + ab\)
- \((a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca\)
These identities provide alternative and faster methods to compute expansions, factorisations and products of expressions.
Using Identities: Solved Examples
Use identities to expand or factor expressions without full multiplication.
Example: Expand \((x + 3)^{2}\).
Solution: Use \((a + b)^{2} = a^{2} + 2ab + b^{2}\) with \(a = x\), \(b = 3\)
Example: Factor \(a^{2} - b^{2}\).
Solution: Use \(a^{2} - b^{2} = (a + b)(a - b)\)
FAQs on Chapter Notes: Algebraic Expressions and Identities
| 1. What are algebraic expressions and how are they different from numerical expressions? |  |
| 2. How do you add and subtract algebraic expressions? | |
| 3. What is the process for multiplying a monomial by a polynomial? | |
| 4. Can you explain what identities are in algebra? | |
| 5. What are standard identities in algebra and provide an example? | |
