NCERT Solutions Class 8 Maths Chapter 9 Exercise 9.4: Algebraic Expressions and Identities

Multiplying Binomials: The Core Concept

The key to multiplying polynomials is the Distributive Law of Multiplication over Addition. When we multiply a binomial (a + b) by another binomial (c + d), we must distribute each individual term of the first binomial over every term of the second binomial.

Mathematically, we write this process as:

(a + b)(c + d) = a × (c + d) + b × (c + d)

Next, we expand this further using basic monomial-binomial multiplication:

= ac + ad + bc + bd

Notice how we multiplied the First terms (a × c), the Outer terms (a × d), the Inner terms (b × c), and the Last terms (b × d). This is often remembered as the FOIL method. After expanding, look out for "like terms" (terms with the exact same variables and exponents) and group them together to simplify the final algebraic expression to its cleanest form.

Step-by-Step Method to Multiply Binomials

1. Write the Expression — Write down the multiplication problem clearly, keeping parentheses around each binomial to keep the terms grouped: (Term1 + Term2) × (Term3 + Term4).
2. Distribute the First Term — Multiply the first term of the first binomial by the entire second binomial: Term1 × (Term3 + Term4).
3. Distribute the Second Term — Multiply the second term of the first binomial by the entire second binomial: Term2 × (Term3 + Term4). Make sure you carry any positive or negative signs carefully.
4. Expand and Multiply — Perform individual multiplications for each pair of monomials. Combine numerical coefficients and add the exponents of like variables.
5. Combine Like Terms — Scan your expanded expression for terms with the same algebraic variables. Add or subtract their coefficients to write the final simplified answer.

Worked Examples from NCERT Exercise 9.4

• Example 1: Multiply (2x + 5) and (4x - 3). Step 1: Write expression: (2x + 5)(4x - 3). Step 2: Distribute: 2x(4x - 3) + 5(4x - 3). Step 3: Expand: (2x × 4x) - (2x × 3) + (5 × 4x) - (5 × 3) = 8x^2 - 6x + 20x - 15. Step 4: Combine like terms (-6x + 20x = 14x): Answer: 8x^2 + 14x - 15.
• Example 2: Multiply (a^2 + 2b^2) and (5a - 3b). Step 1: Write expression: (a^2 + 2b^2)(5a - 3b). Step 2: Distribute: a^2(5a - 3b) + 2b^2(5a - 3b). Step 3: Expand: 5a^3 - 3a^2b + 10ab^2 - 6b^3. Step 4: Check for like terms. Since there are no terms with identical variable factors, the expansion is already simplified. Answer: 5a^3 - 3a^2b + 10ab^2 - 6b^3.

Exam Tips & Common Pitfalls

Always watch the negative signs! A very common mistake in CBSE class 8 algebraic expressions and identities ex 9 4 exams is forgetting that multiplying a negative term by another negative term yields a positive term (e.g., -2 × -3 = +6). Also, when distributing a negative coefficient outside parentheses, like -3(2x - 5), ensure you change all signs inside to get -6x + 15. Always write out the split distributive step before doing the actual multiplication to avoid calculation slips!

Practice Questions with Solutions

• Q: Multiply the binomials: (3/4 a^2 + 3b^2) and 4(a^2 - 2/3 b^2). A: Step 1: Simplify the second expression by distributing the constant 4: 4(a^2 - 2/3 b^2) = 4a^2 - 8/3 b^2. Step 2: Set up the product: (3/4 a^2 + 3b^2)(4a^2 - 8/3 b^2). Step 3: Distribute the terms: 3/4 a^2 × (4a^2 - 8/3 b^2) + 3b^2 × (4a^2 - 8/3 b^2). Step 4: Multiply term by term: = (3/4 a^2 × 4a^2) - (3/4 a^2 × 8/3 b^2) + (3b^2 × 4a^2) - (3b^2 × 8/3 b^2) = 3a^4 - 2a^2b^2 + 12a^2b^2 - 8b^4. Step 5: Combine like terms (-2a^2b^2 + 12a^2b^2 = 10a^2b^2): Final answer: 3a^4 + 10a^2b^2 - 8b^4.
• Q: Simplify the expression: (x + y)(2x + y) + (x + 2y)(x - y). A: Step 1: Multiply the first pair of binomials (x + y)(2x + y): = x(2x + y) + y(2x + y) = 2x^2 + xy + 2xy + y^2 = 2x^2 + 3xy + y^2. Step 2: Multiply the second pair of binomials (x + 2y)(x - y): = x(x - y) + 2y(x - y) = x^2 - xy + 2xy - 2y^2 = x^2 + xy - 2y^2. Step 3: Add both parts together: = (2x^2 + 3xy + y^2) + (x^2 + xy - 2y^2). Step 4: Combine like terms: = (2x^2 + x^2) + (3xy + xy) + (y^2 - 2y^2) = 3x^2 + 4xy - y^2. Final answer: 3x^2 + 4xy - y^2.
• Q: Multiply (2.5l - 0.5m) and (2.5l + 0.5m). A: Step 1: Write down the binomial product: (2.5l - 0.5m)(2.5l + 0.5m). Step 2: Distribute terms: 2.5l(2.5l + 0.5m) - 0.5m(2.5l + 0.5m). Step 3: Multiply the terms out: = (2.5l × 2.5l) + (2.5l × 0.5m) - (0.5m × 2.5l) - (0.5m × 0.5m) = 6.25l^2 + 1.25lm - 1.25lm - 0.25m^2. Step 4: Combine like terms (1.25lm - 1.25lm = 0): Final answer: 6.25l^2 - 0.25m^2.
• Q: Simplify the binomial-trinomial product: (a + c)(a + b - c). A: Step 1: Distribute each term of the binomial over the trinomial: = a(a + b - c) + c(a + b - c). Step 2: Expand the expressions: = a^2 + ab - ac + ca + cb - c^2. Step 3: Re-arrange terms and note that -ac + ca = 0: = a^2 + ab + cb - c^2. Final answer: a^2 + ab + bc - c^2.

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