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Factorisation - Division of Polynomials

Grade 8ICSE

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 applying the Law of Exponents for the variable parts. Visually, imagine a fraction where terms in the numerator and denominator are cancelled out; for any variable xx, we use the rule xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}.

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Division of a Polynomial by a Monomial: To divide a polynomial by a monomial, we divide each term of the polynomial by the given monomial separately. Visually, this can be represented as splitting a single large fraction into several smaller fractions, each sharing the same denominator.

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Factorisation Method for Division: When dividing a polynomial by another polynomial, we first factorise both the dividend and the divisor where possible. If they share common factors, these factors can be cancelled out from both the numerator and the denominator, leaving the simplified quotient.

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Arranging Terms in Standard Form: Before starting long division, both the dividend and the divisor must be written in descending order of their powers (degrees). If a power is missing in the sequence, it is helpful to visualize it with a coefficient of zero, like 0x20x^2, to keep columns aligned.

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Polynomial Long Division Process: This is a repetitive process involving four main steps: Divide (the first term), Multiply, Subtract, and Bring Down. Visually, it looks like a long division 'house' where the dividend is placed inside and the divisor is placed to the left.

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Degree of the Quotient: In polynomial division, the degree of the quotient is equal to the difference between the degree of the dividend and the degree of the divisor. For example, dividing a cubic polynomial (x3x^3) by a linear polynomial (x1x^1) will result in a quadratic quotient (x2x^2).

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The Division Algorithm: Just like in arithmetic, the relationship between the parts of a division can be checked using the formula: Dividend=(Divisor×Quotient)+RemainderDividend = (Divisor \times Quotient) + Remainder. If the remainder is 00, it indicates that the divisor is a factor of the dividend.

📐Formulae

xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}

Dividend=(Divisor×Quotient)+RemainderDividend = (Divisor \times Quotient) + Remainder

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)

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 as a fraction: 24(x2yz+xy2z+xyz2)8xyz\frac{24(x^2yz + xy^2z + xyz^2)}{8xyz}. \ Step 2: Divide the numerical coefficients: 248=3\frac{24}{8} = 3. \ Step 3: Divide each term in the bracket by xyzxyz: \ x2yzxyz=x\frac{x^2yz}{xyz} = x, \ xy2zxyz=y\frac{xy^2z}{xyz} = y, \ xyz2xyz=z\frac{xyz^2}{xyz} = z. \ Step 4: Combine the results: 3(x+y+z)3(x + y + z).

Explanation:

This is a division of a polynomial by a monomial. We factored out the numerical coefficient first and then distributed the division across each term within the parentheses.

Problem 2:

Divide (z2−4z−12)(z^2 - 4z - 12) by (z+2)(z + 2) using the factorisation method.

Solution:

Step 1: Factorise the dividend z2−4z−12z^2 - 4z - 12. We look for two numbers that multiply to −12-12 and add to −4-4. Those numbers are −6-6 and +2+2. \ Step 2: Rewrite the dividend: z2−6z+2z−12=z(z−6)+2(z−6)=(z−6)(z+2)z^2 - 6z + 2z - 12 = z(z - 6) + 2(z - 6) = (z - 6)(z + 2). \ Step 3: Set up the division: (z−6)(z+2)(z+2)\frac{(z - 6)(z + 2)}{(z + 2)}. \ Step 4: Cancel the common factor (z+2)(z + 2) from the numerator and denominator. \ Step 5: The result is (z−6)(z - 6).

Explanation:

The problem is solved by factorising the quadratic trinomial in the numerator. Since (z+2)(z+2) is a factor of the dividend, it cancels out with the divisor, leaving a remainder of zero.