NCERT Solutions for Class 8 Maths Chapter 14 - Factorisation Exercise 14.3

The Art of Division in Algebra: Ex 14.3 Focus

Division is the inverse operation of multiplication. In algebra, when we divide an algebraic expression, we are essentially looking for an expression that, when multiplied by the divisor, gives us the dividend. Exercise 14.3 primarily focuses on the division of algebraic expressions, which often relies heavily on our factorisation skills. We'll explore three main types of division: dividing a monomial by a monomial, a polynomial by a monomial, and a polynomial by another polynomial. Understanding how to correctly cancel common factors is key. Remember, you can only cancel factors, not terms separated by '+' or '-' signs. This skill simplifies complex fractions and equations, making them easier to manage and solve.

Step-by-Step Division of Algebraic Expressions

1. Step 1: Dividing a Monomial by a Monomial — To divide a monomial by another monomial, first divide the numerical coefficients. Then, divide the variable parts separately. For variables with the same base, use the exponent rule: a^m / a^n = a^(m-n). For example, 28x^4 / 7x can be seen as (28/7) * (x^4/x) = 4x^(4-1) = 4x^3.
2. Step 2: Dividing a Polynomial by a Monomial — When dividing a polynomial by a monomial, distribute the division. This means you divide each term of the polynomial (the dividend) by the monomial (the divisor) separately. Then, simplify each resulting monomial division as described in Step 1. For instance, (10x^2 + 5x) / 5x becomes (10x^2 / 5x) + (5x / 5x).
3. Step 3: Dividing a Polynomial by a Polynomial (Using Factorisation) — This is where factorisation truly shines. To divide a polynomial by another polynomial (especially if the divisor is a binomial or trinomial), first factorise the dividend (the polynomial being divided). Look for common factors, or try to apply algebraic identities (like (a+b)^2, (a-b)^2, a^2-b^2). Once factorised, identify and cancel out any common factors present in both the numerator (dividend) and the denominator (divisor).

Solved Examples for Clear Understanding

• Example 1: Divide 28x^4 by 56x. Step 1: Write as a fraction: (28x^4) / (56x). Step 2: Separate numerical and variable parts: (28/56) (x^4/x). Step 3: Simplify numerical part: 28/56 = 1/2. Step 4: Simplify variable part using exponent rule (x^m / x^n = x^(m-n)): x^4 / x^1 = x^(4-1) = x^3. Step 5: Combine results: (1/2) x^3 = x^3 / 2. Final Answer: x^3 / 2.
• Example 2: Divide (24x^2y^3 + 18x^3y^2) by 6xy. Step 1: Divide each term of the polynomial by the monomial: (24x^2y^3 / 6xy) + (18x^3y^2 / 6xy). Step 2: Simplify the first term: (24/6) (x^2/x) (y^3/y) = 4 x^(2-1) y^(3-1) = 4xy^2. Step 3: Simplify the second term: (18/6) (x^3/x) (y^2/y) = 3 x^(3-1) y^(2-1) = 3x^2y. Step 4: Combine the simplified terms: 4xy^2 + 3x^2y. Final Answer: 4xy^2 + 3x^2y.
• Example 3: Divide (y^2 - 16) by (y + 4). Step 1: Factorise the numerator (dividend). Recognize y^2 - 16 as a difference of squares (a^2 - b^2 = (a-b)(a+b)). Here a=y, b=4. So, y^2 - 16 = (y - 4)(y + 4). Step 2: Rewrite the division with the factorised numerator: ((y - 4)(y + 4)) / (y + 4). Step 3: Cancel the common factor (y + 4) from numerator and denominator. Step 4: The remaining expression is (y - 4). Final Answer: y - 4.

Important Tip: Common Division Errors to Avoid

When dividing algebraic expressions, students often make a few common mistakes that can lead to incorrect answers. One major error is incorrect cancellation. Remember, you can only cancel out common factors, not terms. For example, in (x+y)/x, you cannot cancel 'x' to get 'y' because 'x' in the numerator is part of a sum, not a standalone factor. Another frequent mistake is sign errors, especially when distributing division to negative terms. Always pay close attention to the signs. Lastly, forgetting to factorise completely before cancelling common polynomial factors is a pitfall. Make sure the dividend is fully factorised to reveal all possible common factors with the divisor.

Practice Questions with Solutions

• Q: Divide -72x^3y^2z by 12xy. A: Step 1: Write the division as a fraction: (-72x^3y^2z) / (12xy). Step 2: Separate numerical and variable parts: (-72/12) (x^3/x) (y^2/y) z. Step 3: Simplify numerical part: -72/12 = -6. Step 4: Simplify variable parts using exponent rule: x^3/x = x^(3-1) = x^2, y^2/y = y^(2-1) = y^1 = y. The 'z' remains as is. Step 5: Combine results: -6 x^2 y z = -6x^2yz. Final answer: -6x^2yz.
• Q: Divide (15a^4b^3 - 9a^3b^2 + 3a^2b) by 3a^2b. A: Step 1: Divide each term of the polynomial by the monomial: (15a^4b^3 / 3a^2b) - (9a^3b^2 / 3a^2b) + (3a^2b / 3a^2b). Step 2: Simplify the first term: (15/3) (a^4/a^2) (b^3/b) = 5 a^(4-2) b^(3-1) = 5a^2b^2. Step 3: Simplify the second term: (9/3) (a^3/a^2) (b^2/b) = 3 a^(3-2) b^(2-1) = 3ab. Step 4: Simplify the third term: (3/3) (a^2/a^2) (b/b) = 1 a^0 b^0 = 1 1 1 = 1. Step 5: Combine the simplified terms: 5a^2b^2 - 3ab + 1. Final answer: 5a^2b^2 - 3ab + 1.
• Q: Divide (x^2 + 7x + 10) by (x + 2). A: Step 1: Factorise the quadratic expression in the numerator (dividend). We need two numbers that multiply to 10 and add up to 7. These numbers are 5 and 2. So, x^2 + 7x + 10 = (x + 5)(x + 2). Step 2: Rewrite the division with the factorised numerator: ((x + 5)(x + 2)) / (x + 2). Step 3: Cancel the common factor (x + 2) from numerator and denominator. Step 4: The remaining expression is (x + 5). Final answer: x + 5.
• Q: Divide (49p^2 - 36q^2) by (7p - 6q). A: Step 1: Factorise the numerator (dividend). Recognize 49p^2 - 36q^2 as a difference of squares (a^2 - b^2 = (a-b)(a+b)). Here a = 7p and b = 6q. So, 49p^2 - 36q^2 = (7p - 6q)(7p + 6q). Step 2: Rewrite the division with the factorised numerator: ((7p - 6q)(7p + 6q)) / (7p - 6q). Step 3: Cancel the common factor (7p - 6q) from numerator and denominator. Step 4: The remaining expression is (7p + 6q). Final answer: 7p + 6q.

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