Topics to be learn : Introduction to polynomial Degree of a polynomial To divide a monomial by a monomial To divide a polynomial by a monomial To divide a polynomial by a binomial

Recall :

Addition, subtraction and multiplication on algebraic expressions :

(1) 2a + 3a = 5a

(2) 7b − 4b = 3b

(3) 3p × p² = 3p³

(4) 5m² × 3m² = 15m⁴

(5) (2x + 5y) × \(\frac{3}{x}\) = 6 + \(\frac{15y}{x}\)

(6) (3x² + 4y) × (2x + 3y) = 6x³ + 9x²y + 8xy + 12y²

Introduction to polynomial :

• To perform operations on polynomials, one must first understand their structural properties.
• Polynomials are specific types of algebraic expressions that follow strict rules regarding their variables and indices.

Polynomial Definition: An algebraic expression in one variable is considered a polynomial if the index (exponent) of every term is a whole number.

Examples:

(1) x² + 2x + 3 : is a polynomial because all indices are whole numbers.

Here, the powers or indices of the variable are 2. 1, 0, (3 means 3x⁰).

(2) 3y³ + 2y² + y + 5 : is a polynomial because all indices are whole numbers.

Here, the powers or indices of the variable are 3, 2. 1, 0, (5 means 5x⁰).

(3) x² + \(\frac{2}{x}\)  + 3 : is not a polynomial.

Here, \(\frac{2}{x}\) means 2x⁻¹. The index of \(\frac{2}{x}\) is −1 which is negative and not a whole number.

Degree of a Polynomial:

This is defined as the greatest index of the variable present in the given polynomial.

Examples :

(1) In 3x² + 4x, the greatest index is 2; therefore, the degree is 2.

(2) In 7x³ + 5x + 4x⁵ + 2x², the greatest index is 5; therefore, the degree is 5.

Operations such as addition, subtraction, and multiplication on polynomials are performed similarly to standard algebraic expressions.

To divide a monomial by a monomial :

Examples :

(1) Divide : 15p³ ÷ 3p

Solution:

Division is the opposite operation of multiplication.

For division 15p³ ÷ 3p, we find the multiplier of 3p which will give product 15p³.

3p × 5p² = 15p³

∴ 15p³ ÷ 3p = 5p²

(2) Divide and write the correct terms in the boxes.

(i) (−36x⁴) ÷ (−9x)

(ii) (5m²) ÷ (−m)

 

(iii) (−20y⁵) ÷ (2y³)

To divide a polynomial by a monomial :

Study the following examples

(1) (6x³ + 8x²) ÷ 2x

Solution :

Explanation −

(i) 2x × 3x² = 6x³

(ii) 2x × 4x = 8x²

∴ Quotient = 3x² + 4x

Remainder = 0

 

 (2) (15y⁴ + 10y³ − 3y²) ÷ 5y²

Solution :

Explanation −

(i) 5y² × 3y² = 15y⁴

(ii) 5y² × 2y = 10y³

(iii) 5y² × \(\frac{3}{5}\) = −3y²

∴ Quotient = 3y² + 2y − \(\frac{3}{5}\),  Remainder = 0

 Note : While dividing a polynomial, the operation of division ends when either the remainder is zero or the degree of the remainder is less than the degree of the divisior polynomial.

To divide a polynomial by a binomial :

The method of division of a polynomial by a binomial is the same as the division of a polynomial by a monomial.

Study the following examples :

(1) (y⁴ + 24y − 10y² ) ÷ (y + 4)

Solution:

• In this example, degree of the dividend polynomial is 4.
• The indices of variable in it are not in descending order.
• The term with index 3 is missing. Assume it as 0y³.
• Write the dividend in the descending order of indices and then divide.

Explanation −

(i) (y + 4) × y³ = y⁴+ 4y³

(ii) (y + 4) × −4y² = −4y³ − 16y²

(iii) (y + 4) × 6y = 6y² + 24y

∴ Quotient = y³ − 4y² + 6y ; Remainder = 0

 

Procedural Standards and Rules :

For accurate computation, specific structural rules must be followed during the setup and execution of the division.

• Descending Order: The terms of both the dividend and the divisor must be written in descending order of their indices before starting the division.
• Missing Terms: if a term with a specific index is missing from the sequence (e.g., a y³ term is missing in a polynomial starting with y⁴), it must be included with a coefficient of zero (e.g. 0y³). This ensures the alignment of terms during subtraction.

Criteria for Termination :

• The operation of division is considered complete only when one of the following two conditions is met:
• The remainder is zero.
• The degree of the remainder polynomial is strictly less than the degree of the divisor polynomial.