Multiplication of Algebraic Expressions
In earlier classes, you learnt to add and subtract algebraic expressions. Multiplication of algebraic expressions is the next fundamental operation in algebra.
Multiplication of algebraic expressions follows the distributive property: each term of one expression is multiplied with every term of the other expression, and the results are added.
In Class 8 NCERT Maths, you learn to multiply a monomial by a monomial, a monomial by a polynomial, and a polynomial by a polynomial (especially binomial × binomial and binomial × trinomial).
This skill is essential for expanding algebraic identities, factorising expressions, solving equations, and working with formulas in geometry and physics.
What is Multiplication of Algebraic Expressions?
Key terms:
- Monomial — an expression with one term (e.g., 3x, −5y², 7ab).
- Binomial — an expression with two terms (e.g., 2x + 3, a − b).
- Trinomial — an expression with three terms (e.g., x² + 2x + 1).
- Polynomial — an expression with one or more terms.
Rules for multiplying terms:
- Multiply the coefficients (numerical parts).
- Multiply the variables using the law of exponents: xᵐ × xⁿ = xᵐ⁺ⁿ.
- Apply the sign rules: (+)(+) = +, (−)(−) = +, (+)(−) = −, (−)(+) = −.
Multiplication of Algebraic Expressions Formula
Monomial × Monomial:
(a · xᵐ) × (b · xⁿ) = ab · xᵐ⁺ⁿ
Monomial × Binomial (Distributive Property):
a(b + c) = ab + ac
Binomial × Binomial:
(a + b)(c + d) = ac + ad + bc + bd
Laws of exponents used:
- xᵐ × xⁿ = xᵐ⁺ⁿ
- (xᵐ)ⁿ = xᵐⁿ
- x⁰ = 1
Derivation and Proof
Why the distributive property works:
The distributive property states that multiplying a sum by a number is the same as multiplying each addend by that number and then adding:
- a × (b + c) = a × b + a × c
Extending to binomial × binomial:
- (a + b)(c + d)
- = a(c + d) + b(c + d) — treating (c + d) as a single quantity
- = ac + ad + bc + bd — applying distributive property again
Extending to binomial × trinomial:
- (a + b)(c + d + e)
- = a(c + d + e) + b(c + d + e)
- = ac + ad + ae + bc + bd + be
Each term of the first expression multiplies every term of the second. If the first has m terms and the second has n terms, the product initially has m × n terms (before combining like terms).
Types and Properties
Types of multiplication problems:
1. Monomial × Monomial:
- Example: 3x² × 5x³ = 15x⁵
2. Monomial × Binomial:
- Example: 2x(3x + 4) = 6x² + 8x
3. Monomial × Trinomial:
- Example: 3a(a² + 2a − 5) = 3a³ + 6a² − 15a
4. Binomial × Binomial:
- Example: (x + 3)(x + 5) = x² + 8x + 15
5. Binomial × Trinomial:
- Example: (x + 2)(x² + 3x + 1)
6. Using identities for quick multiplication:
- (a + b)², (a − b)², (a + b)(a − b) are special cases.
Solved Examples
Example 1: Example 1: Monomial × Monomial
Problem: Multiply 4x³ by 3x².
Solution:
- Coefficients: 4 × 3 = 12
- Variables: x³ × x² = x³⁺² = x⁵
- Result = 12x⁵
Answer: 4x³ × 3x² = 12x⁵.
Example 2: Example 2: Monomial × Monomial with two variables
Problem: Multiply (−2a²b) by (5ab³).
Solution:
- Coefficients: (−2) × 5 = −10
- Variables: a² × a = a³; b × b³ = b⁴
- Result = −10a³b⁴
Answer: (−2a²b)(5ab³) = −10a³b⁴.
Example 3: Example 3: Monomial × Binomial
Problem: Multiply 3x by (2x + 5).
Solution:
- 3x × (2x + 5) = 3x × 2x + 3x × 5
- = 6x² + 15x
Answer: 3x(2x + 5) = 6x² + 15x.
Example 4: Example 4: Monomial × Trinomial
Problem: Multiply −4y by (y² − 3y + 7).
Solution:
- −4y × y² = −4y³
- −4y × (−3y) = +12y²
- −4y × 7 = −28y
- Result = −4y³ + 12y² − 28y
Answer: −4y(y² − 3y + 7) = −4y³ + 12y² − 28y.
Example 5: Example 5: Binomial × Binomial
Problem: Multiply (x + 4)(x + 7).
Solution:
- x × x = x²
- x × 7 = 7x
- 4 × x = 4x
- 4 × 7 = 28
- Result = x² + 7x + 4x + 28 = x² + 11x + 28
Answer: (x + 4)(x + 7) = x² + 11x + 28.
Example 6: Example 6: Binomial × Binomial with subtraction
Problem: Multiply (2a − 3)(4a + 5).
Solution:
- 2a × 4a = 8a²
- 2a × 5 = 10a
- (−3) × 4a = −12a
- (−3) × 5 = −15
- Result = 8a² + 10a − 12a − 15 = 8a² − 2a − 15
Answer: (2a − 3)(4a + 5) = 8a² − 2a − 15.
Example 7: Example 7: Binomial × Trinomial
Problem: Multiply (x + 2)(x² + 3x + 5).
Solution:
- x × x² = x³
- x × 3x = 3x²
- x × 5 = 5x
- 2 × x² = 2x²
- 2 × 3x = 6x
- 2 × 5 = 10
- Result = x³ + 3x² + 5x + 2x² + 6x + 10
- = x³ + 5x² + 11x + 10
Answer: (x + 2)(x² + 3x + 5) = x³ + 5x² + 11x + 10.
Example 8: Example 8: Product of three monomials
Problem: Find the product: 2x × 3y × 4z.
Solution:
- Coefficients: 2 × 3 × 4 = 24
- Variables: x × y × z = xyz
- Result = 24xyz
Answer: 2x × 3y × 4z = 24xyz.
Example 9: Example 9: Squaring a binomial using multiplication
Problem: Find (3x + 2)² by actual multiplication.
Solution:
- (3x + 2)² = (3x + 2)(3x + 2)
- 3x × 3x = 9x²
- 3x × 2 = 6x
- 2 × 3x = 6x
- 2 × 2 = 4
- Result = 9x² + 6x + 6x + 4 = 9x² + 12x + 4
Verification using identity (a+b)² = a² + 2ab + b²:
- (3x)² + 2(3x)(2) + 2² = 9x² + 12x + 4 ✓
Answer: (3x + 2)² = 9x² + 12x + 4.
Example 10: Example 10: Finding area using multiplication
Problem: A rectangle has length (2x + 3) cm and breadth (x + 5) cm. Find its area as an algebraic expression.
Solution:
- Area = length × breadth = (2x + 3)(x + 5)
- 2x × x = 2x²
- 2x × 5 = 10x
- 3 × x = 3x
- 3 × 5 = 15
- Area = 2x² + 10x + 3x + 15 = (2x² + 13x + 15) cm²
Answer: The area is (2x² + 13x + 15) cm².
Real-World Applications
Algebraic Identities: The four standard identities of Class 8 are derived by multiplying binomials. For example, (a + b)(a − b) = a² − b².
Area and Volume: When dimensions of geometric shapes are given as algebraic expressions, finding area/volume requires multiplication. For example, area of a rectangle with sides (x + 3) and (x + 5) = x² + 8x + 15.
Solving Equations: Expanding products is a step in solving many equations. For instance, expanding (x + 2)(x + 3) = 20 gives x² + 5x + 6 = 20, a quadratic equation.
Physics Formulas: Kinetic energy = ½ mv², force equations, and other physics formulas require algebraic multiplication for derivation and simplification.
Computer Science: Polynomial multiplication is used in algorithms for signal processing, error-correcting codes, and computational geometry.
Key Points to Remember
- Multiply coefficients and apply the law of exponents for variables: xᵐ × xⁿ = xᵐ⁺ⁿ.
- Follow sign rules: like signs give (+), unlike signs give (−).
- Distributive property: a(b + c) = ab + ac.
- Binomial × Binomial: multiply each term of the first by each term of the second (4 products), then combine like terms.
- If the first expression has m terms and the second has n terms, the product initially has m × n terms.
- Always combine like terms after multiplication.
- Multiplication of expressions is commutative: AB = BA.
- Multiplication is associative: (AB)C = A(BC).
- The degree of the product = sum of degrees of the factors.
- Special products (identities) are shortcuts for frequently occurring multiplications.
Practice Problems
- Multiply 5x²y by 3xy³.
- Expand: 4a(2a² − 3a + 7).
- Multiply (x + 6)(x − 4).
- Multiply (3p − 2q)(2p + 5q).
- Expand (y − 3)(y² + 2y − 5).
- Find the product of (−2x²)(3x)(4x³).
- A square has side (2a + b). Find its area by multiplication.
- Verify that (x + 1)(x + 2)(x + 3) = x³ + 6x² + 11x + 6.
Frequently Asked Questions
Q1. What is the distributive property?
The distributive property states that a(b + c) = ab + ac. It allows us to multiply a term with every term inside the bracket.
Q2. How do you multiply two monomials?
Multiply the coefficients and add the exponents of like variables. For example, 3x² × 4x³ = 12x⁵.
Q3. How do you multiply two binomials?
Multiply each term of the first binomial by each term of the second. This gives 4 terms. Then combine like terms. For example, (x+2)(x+3) = x² + 3x + 2x + 6 = x² + 5x + 6.
Q4. What is the FOIL method?
FOIL stands for First, Outer, Inner, Last. It is a mnemonic for multiplying two binomials: First terms, Outer terms, Inner terms, Last terms.
Q5. What happens when you multiply a positive and a negative term?
The result is negative. For example, 3x × (−4y) = −12xy.
Q6. What is the degree of the product?
The degree of the product equals the sum of the degrees of the two expressions. For example, (x²)(x³) has degree 5.
Q7. How do you multiply a binomial by a trinomial?
Multiply each term of the binomial by each term of the trinomial. This gives 2 × 3 = 6 terms. Combine like terms to simplify.
Q8. Is multiplication of expressions commutative?
Yes. The order does not matter: (x+2)(x+3) = (x+3)(x+2).
Q9. What is the difference between expanding and factorising?
Expanding means multiplying out brackets to get individual terms. Factorising is the reverse — writing an expression as a product of factors.
Q10. Why is this topic important?
Multiplication of algebraic expressions is the foundation for identities, factorisation, solving equations, and working with formulas in geometry, physics, and higher mathematics.
