Algebraic Expressions and Identities Ex 9.3 Class 8 NCERT

The Core Concept: Using the Distributive Property

Multiplying a monomial by a polynomial might sound complicated, but it's based on one simple idea: the distributive property. Think of it like giving toffees to your friends. If you have 3 toffees to give to a group of friends, say (Ram + Shyam), you must give 3 toffees to Ram AND 3 toffees to Shyam. You distribute them fairly to everyone in the group.

In algebra, it works the same way. The monomial outside the bracket is 'distributed' to each term inside the bracket. The rule is:
a × (b + c) = (a × b) + (a × c)

Here, the monomial a is multiplied by the first term b, and then it's multiplied by the second term c. Let's see it with an example. To find the product of 2x and (3y + 5z):

1. Distribute 2x to the first term, 3y. This gives 2x * 3y = 6xy.
2. Distribute 2x to the second term, 5z. This gives 2x * 5z = 10xz.
3. Combine the results: 6xy + 10xz. So, 2x * (3y + 5z) = 6xy + 10xz. This principle works even if the polynomial has three or more terms.

How to Multiply: A Step-by-Step Guide

1. Step 1: Identify the Terms — Look at your problem. Identify the monomial (the single term outside the brackets) and the polynomial (the expression with two or more terms inside the brackets).
2. Step 2: Apply the Distributive Property — Multiply the monomial by the first term of the polynomial. Remember to multiply the numerical coefficients and add the powers of like variables (e.g., p¹ * p² = p¹⁺² = p³).
3. Step 3: Repeat for All Terms — Now, multiply the same monomial by the second term of the polynomial. Pay close attention to the signs (+ or -). If the polynomial has more terms (like a trinomial), repeat this step for every single term inside the brackets.
4. Step 4: Combine and Write the Final Expression — Write down the results from each multiplication, connected by their correct signs. This new expression is your final answer. There's no need to simplify further unless there are like terms.

Worked Examples from Ex 9.3

• Problem: Find the product: 5a (6a - 7b) Solution: 1. Distribute 5a to the first term 6a: 5a 6a = (5 6) (a a) = 30a² 2. Distribute 5a to the second term -7b: 5a (-7b) = (5 -7) (a b) = -35ab 3. Combine the results: 30a² - 35ab Final Answer: 5a (6a - 7b) = 30a² - 35ab
• Problem: Find the product: 4p² (p² - q²) Solution: 1. Distribute 4p² to the first term p²: 4p² p² = 4 (p² p²) = 4p⁴ (Remember to add the exponents: 2+2=4) 2. Distribute 4p² to the second term -q²: 4p² (-q²) = -4 (p² q²) = -4p²q² 3. Combine the results: 4p⁴ - 4p²q² Final Answer: 4p² (p² - q²) = 4p⁴ - 4p²q²
• Problem: Simplify x(x - 3) + 2 and find its value for x = 2. Solution: 1. First, simplify the expression by multiplying: x(x - 3) + 2 = (x x) - (x 3) + 2 = x² - 3x + 2 2. Now, substitute the value x = 2 into the simplified expression: Value = (2)² - 3(2) + 2 3. Calculate the result: 4 - 6 + 2 = -2 + 2 = 0 Final Answer: The simplified expression is x² - 3x + 2, and its value at x=2 is 0.

Common Mistakes to Avoid

• Incomplete Distribution: A very common error is to multiply the monomial only with the first term in the bracket. Always remember to distribute it to every term. For example, 3(x + 4) is 3x + 12, not 3x + 4.
• Sign Errors: Be very careful with negative signs. When multiplying a positive term by a negative term, the result is negative. When multiplying a negative term by another negative term, the result becomes positive. For example, -2a * (b - 3c) = -2ab + 6ac.
• Incorrectly Handling Exponents: Remember the rule of exponents: a^m a^n = a^(m+n). You must add the powers of the same variable, not multiply them. For example, y² y³ = y⁵, not y⁶.
• Forgetting Coefficients: Don't forget to multiply the numbers (coefficients) in front of the variables. In (4x) * (2x² + 3), the first part is 8x³, not 4x³.

Practice Questions with Solutions

• Q: Carry out the multiplication of the expression 7pqr (p - q + r). A: Step 1: Identify the monomial 7pqr and the trinomial (p - q + r). Step 2: Multiply 7pqr with the first term p: 7pqr p = 7p²qr. Step 3: Multiply 7pqr with the second term -q: 7pqr (-q) = -7pq²r. Step 4: Multiply 7pqr with the third term r: 7pqr r = 7pqr². Step 5: Combine the results. Final answer: 7p²qr - 7pq²r + 7pqr²
• Q: Find the product: (2/3)xy (9x²y - 12xy²). A: Step 1: Apply the distributive property. Step 2: Multiply (2/3)xy with the first term 9x²y: (2/3 9) (x x²) (y y) = 6x³y². Step 3: Multiply (2/3)xy with the second term -12xy²: (2/3 -12) (x x) (y * y²) = -8x²y³. Step 4: Combine the products. Final answer: 6x³y² - 8x²y³
• Q: Simplify 3y(2y - 7) - 3(y - 4) - 63 and find its value for y = -2. A: Step 1: First, expand the expressions. 3y(2y - 7) = 6y² - 21y. And -3(y - 4) = -3y + 12. Step 2: Combine all parts: 6y² - 21y - 3y + 12 - 63. Step 3: Combine like terms (-21y - 3y and +12 - 63): 6y² - 24y - 51. Step 4: Substitute y = -2 into the simplified expression: 6(-2)² - 24(-2) - 51. Step 5: Calculate the value: 6(4) + 48 - 51 = 24 + 48 - 51 = 72 - 51 = 21. Final answer: The simplified expression is 6y² - 24y - 51, and its value for y = -2 is 21.
• Q: Add the products: a(a - b) and b(b - a). A: Step 1: Find the first product: a(a - b) = a² - ab. Step 2: Find the second product: b(b - a) = b² - ba. Note that ba is the same as ab. Step 3: Add the two resulting expressions: (a² - ab) + (b² - ab). Step 4: Combine the like terms (-ab and -ab): a² + b² - 2ab. Final answer: a² - 2ab + b²

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