CBSE Class 8 Maths: Algebraic Expressions and Identities Ex 9.2

Understanding the Building Blocks: Monomials and Polynomials

Before we jump into multiplication, let's quickly review what algebraic expressions are made of. An algebraic expression is a combination of variables (like x, y, a, b), constants (like 2, -5, 1/2), and mathematical operations (+, -, *, /).

• Term: Each part of an expression separated by a plus (+) or minus (-) sign is called a term. For example, in 3x^2 + 5y - 7, 3x^2, 5y, and 7 are terms.
• Coefficient: The numerical part of a term is its coefficient. In 3x^2, 3 is the coefficient. In y, the coefficient is 1.
• Variable: The literal (letter) part of a term that can take different values. Example: x, y.

Now, let's categorize expressions based on the number of terms:

• Monomial: An expression with one term. Examples: 5x, -3y^2, 7, 2ab.
• Binomial: An expression with two terms. Examples: 2x + 3, a^2 - 4b, pq + 7.
• Trinomial: An expression with three terms. Examples: x^2 + 2x - 1, 4a + 5b - 2c.
• Polynomial: A general name for an expression with one or more terms. So, monomials, binomials, and trinomials are all types of polynomials. In Exercise 9.2, you'll mainly work with multiplying monomials by other monomials or by polynomials (which could be binomials or trinomials).

Mastering Monomial Multiplication

1. Step 1: Multiply the Numerical Coefficients — Identify the numerical parts (coefficients) of both monomials and multiply them together. Remember to pay attention to their signs (positive or negative).
2. Step 2: Multiply the Variable Parts — For each variable, multiply its corresponding parts. If the same variable appears in both monomials (e.g., x and x^2), you add their exponents. Remember the rule: a^m × a^n = a^(m+n). If a variable appears in only one monomial, it just carries over.
3. Step 3: Combine the Results — Write the product of the numerical coefficients followed by the product of the variable parts. This gives you the final monomial product.

Distributing the Power: Monomial × Polynomial

1. Step 1: Identify the Monomial and Polynomial — Clearly separate the single-term expression (monomial) from the multi-term expression (polynomial) that you need to multiply.
2. Step 2: Apply the Distributive Property — Multiply the monomial by each individual term inside the polynomial. This means you will perform a series of monomial-by-monomial multiplications, as explained in the previous section. For example, A × (B + C) becomes (A × B) + (A × C).
3. Step 3: Simplify and Combine Like Terms (if any) — After multiplying, you will have several terms. Check if there are any 'like terms' (terms with the exact same variable parts and exponents). If there are, combine them by adding or subtracting their coefficients. This simplifies your final answer.

Worked Examples: Applying the Multiplication Rules

• Example 1: Multiply (4x) by (7xy) Step 1: Multiply coefficients: 4 × 7 = 28 Step 2: Multiply variables: x × xy = x^(1+1)y = x^2y Step 3: Combine: 28x^2y Final Answer: 28x^2y
• Example 2: Find the product of (-3a^2b) and (5ab^3) Step 1: Multiply coefficients: (-3) × 5 = -15 Step 2: Multiply variables: a^2b × ab^3 = a^(2+1)b^(1+3) = a^3b^4 Step 3: Combine: -15a^3b^4 Final Answer: -15a^3b^4
• Example 3: Multiply 2x by (3x + 5y) Step 1: Apply distributive property: 2x × (3x + 5y) = (2x × 3x) + (2x × 5y) Step 2: Perform monomial multiplications: 2x × 3x = (2 × 3) × (x × x) = 6x^2 2x × 5y = (2 × 5) × (x × y) = 10xy Step 3: Combine the results: 6x^2 + 10xy Final Answer: 6x^2 + 10xy
• Example 4: Simplify -4m^2(2m^3 - 3n + 5) Step 1: Apply distributive property: -4m^2 × (2m^3 - 3n + 5) = (-4m^2 × 2m^3) + (-4m^2 × -3n) + (-4m^2 × 5) Step 2: Perform monomial multiplications: -4m^2 × 2m^3 = -8m^(2+3) = -8m^5 -4m^2 × -3n = (-4 × -3) × (m^2 × n) = 12m^2n -4m^2 × 5 = (-4 × 5) × m^2 = -20m^2 Step 3: Combine the results: -8m^5 + 12m^2n - 20m^2 Final Answer: -8m^5 + 12m^2n - 20m^2

Exam Tip: Avoid These Common Pitfalls!

When solving problems from Algebraic Expressions and Identities Ex 9.2, students often make these common mistakes:

1. Forgetting Negative Signs: A negative sign can completely change your answer. Always multiply signs first: (-) (+) = (-), (-) (-) = (+). For example, (-2x) * (3y) = -6xy, not 6xy.
2. Incorrectly Adding Exponents: Remember, when multiplying variables with the same base, you add their powers. x^2 * x^3 = x^(2+3) = x^5. A common mistake is to multiply the exponents (x^6) or simply write x^2x^3 without combining.
3. Missing a Term in Distribution: When a monomial multiplies a polynomial (like a(b+c+d)), make sure the monomial multiplies every single term inside the bracket. A common error is only multiplying the first term, leading to ab+c+d instead of ab+ac+ad.
4. Combining Unlike Terms: After multiplying, you might end up with several terms. Only combine 'like terms' (terms with identical variable parts and powers). For example, 5x^2 + 3x cannot be combined, but 5x^2 + 3x^2 can be combined to 8x^2.

Practice Questions with Solutions

• Q: Find the product of (5pq) and (-7p^2qr). A: Step 1: Multiply the numerical coefficients: 5 × (-7) = -35. Step 2: Multiply the variable parts: pq × p^2qr = p^(1+2)q^(1+1)r = p^3q^2r. Step 3: Combine the results: -35p^3q^2r. Final answer: -35p^3q^2r
• Q: Multiply (3x^2) by (4x + 2y - 5). A: Step 1: Apply the distributive property, multiplying 3x^2 by each term inside the bracket: (3x^2 × 4x) + (3x^2 × 2y) + (3x^2 × -5). Step 2: Perform each monomial multiplication: 3x^2 × 4x = (3 × 4) × (x^2 × x) = 12x^(2+1) = 12x^3. 3x^2 × 2y = (3 × 2) × (x^2 × y) = 6x^2y. 3x^2 × -5 = (3 × -5) × x^2 = -15x^2. Step 3: Combine the results: 12x^3 + 6x^2y - 15x^2. Final answer: 12x^3 + 6x^2y - 15x^2
• Q: Simplify (1/2)ab * (6a^2b - 4ab^2). A: Step 1: Apply the distributive property: ((1/2)ab × 6a^2b) - ((1/2)ab × 4ab^2). Step 2: Perform each monomial multiplication: (1/2)ab × 6a^2b = (1/2 × 6) × (a × a^2) × (b × b) = 3a^(1+2)b^(1+1) = 3a^3b^2. (1/2)ab × 4ab^2 = (1/2 × 4) × (a × a) × (b × b^2) = 2a^(1+1)b^(1+2) = 2a^2b^3. Step 3: Combine the results: 3a^3b^2 - 2a^2b^3. Final answer: 3a^3b^2 - 2a^2b^3
• Q: Calculate the value of (-5x)(x^2 - 2x + 7). A: Step 1: Apply the distributive property, multiplying -5x by each term: (-5x × x^2) + (-5x × -2x) + (-5x × 7). Step 2: Perform each monomial multiplication: -5x × x^2 = -5x^(1+2) = -5x^3. -5x × -2x = (-5 × -2) × (x × x) = 10x^(1+1) = 10x^2. -5x × 7 = -35x. Step 3: Combine the results: -5x^3 + 10x^2 - 35x. Final answer: -5x^3 + 10x^2 - 35x

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