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Factorisation - Division of Algebraic Expressions

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Division of a Monomial by another Monomial: This involves dividing the numerical coefficients and then using the laws of exponents to simplify the variables. Imagine the division as a fraction where you cross out common factors from the top (numerator) and bottom (denominator).

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Division of a Polynomial by a Monomial: Each term of the polynomial is divided individually by the monomial. You can visualize this as splitting a single large fraction into several smaller fractions, each having the same denominator.

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Division of a Polynomial by a Polynomial: The most effective way is to factorise both the numerator and the denominator completely and then cancel out the common factors. Think of it as breaking down complex blocks into simpler, identical units that can be removed.

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Factorisation via Regrouping: Sometimes, terms must be rearranged or grouped in pairs to find a common factor before division can occur. Visualise this as rearranging scattered pieces to find matching pairs.

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Using Algebraic Identities: Many division problems are simplified by identifying patterns like the difference of two squares a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b). Visualise the dividend as an area that can be reshaped into a product of a length and a breadth.

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The Cancellation Rule: Only factors that are multiplied together can be cancelled. You cannot cancel individual terms that are separated by addition or subtraction signs. For example, in x+5x\frac{x+5}{x}, you cannot cancel the xx.

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Law of Exponents in Division: When dividing same bases, we subtract the exponents: xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}. Visualise a stack of variables where the bottom ones 'neutralise' an equal number of variables on top.

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Verification of Division: The result of a division can always be checked by multiplying the quotient by the divisor. If the product equals the dividend, the division is correct: Dividend=Divisor×Quotient\text{Dividend} = \text{Divisor} \times \text{Quotient}.

📐Formulae

Dividend=Divisor×Quotient\text{Dividend} = \text{Divisor} \times \text{Quotient}

xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}

a+b+cm=am+bm+cm\frac{a+b+c}{m} = \frac{a}{m} + \frac{b}{m} + \frac{c}{m}

a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2

a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2

x2+(a+b)x+ab=(x+a)(x+b)x^2 + (a + b)x + ab = (x + a)(x + b)

💡Examples

Problem 1:

Divide 24(x2yz+xy2z+xyz2)24(x^2yz + xy^2z + xyz^2) by 8xyz8xyz.

Solution:

Step 1: Write the expression in fraction form: 24(x2yz+xy2z+xyz2)8xyz\frac{24(x^2yz + xy^2z + xyz^2)}{8xyz} Step 2: Factor out the common term xyzxyz from the bracket in the numerator: 24xyz(x+y+z)8xyz\frac{24xyz(x + y + z)}{8xyz} Step 3: Cancel the common factors xyzxyz and divide the coefficients 24÷8=324 \div 8 = 3: 3(x+y+z)3(x + y + z) Final Answer: 3x+3y+3z3x + 3y + 3z or 3(x+y+z)3(x + y + z).

Explanation:

This problem demonstrates dividing a polynomial by a monomial by first identifying a common factor in the numerator to simplify the division.

Problem 2:

Divide (z2−4z−12)(z^2 - 4z - 12) by (z+2)(z + 2).

Solution:

Step 1: Factorise the numerator z2−4z−12z^2 - 4z - 12 using the splitting the middle term method. We need two numbers that multiply to −12-12 and add to −4-4. Those numbers are −6-6 and 22. z2−6z+2z−12z^2 - 6z + 2z - 12 z(z−6)+2(z−6)z(z - 6) + 2(z - 6) (z−6)(z+2)(z - 6)(z + 2) Step 2: Set up the division: (z−6)(z+2)(z+2)\frac{(z - 6)(z + 2)}{(z + 2)} Step 3: Cancel the common binomial factor (z+2)(z + 2) from both the numerator and the denominator: z−6z - 6 Final Answer: z−6z - 6.

Explanation:

This shows how to divide a polynomial by another polynomial by factorising the quadratic expression in the dividend first.