Maharashtra Board Class 8 Maths part 2 Chapter 10 Division of Polynomials PDF Download

Division of Polynomials

Let's Recall

Last year we have studied how to perform addition, subtraction and multiplication on algebraic expressions.

Fill in the blanks in the following examples.

(1) 2a + 3a =

(2) 7b - 4b =

(3) 3p × p² =

(4) 5m² × 3m² =

(5) (2x + 5y) × \(\frac{3}{x}\) =

(6) (3x² + 4y) × (2x + 3y) =

Introduction To Polynomial

If index of each term of an algebraic expression in one variable is a whole number, then the expression is called a polynomial.

For example, \(x^2 + 2x + 3\) and \(3y^3 + 2y^2 + y + 5\) are polynomials in one variable.

Polynomials are specific algebraic expressions. Hence the operations of addition, subtraction and multiplication on polynomials are similar to those performed on algebraic expressions.

Ex. (1) \((3x^2 - 2x) \times (4x^3 - 3x^2)\)

\(= 3x^2(4x^3 - 3x^2) - 2x(4x^3 - 3x^2)\)

\(= 12x^5 - 9x^4 - 8x^4 + 6x^3\)

\(= 12x^5 - 17x^4 + 6x^3\)

Ex. (2) \((4x - 5) - (3x^2 - 7x + 8)\)

\(= 4x - 5 - 3x^2 + 7x - 8\)

\(= -3x^2 + 4x + 7x - 8 - 5\)

\(= -3x^2 + 11x - 13\)

Teacher's Note

A polynomial is just an expression made of variables and numbers. Think of it like words in a sentence - each word is important and has its place, just like each term in a polynomial.

Exam Trick

Remember: The degree of a polynomial is the HIGHEST power (index) of the variable. Write all terms in order from biggest power to smallest power - this helps you find the degree quickly.

Points to Remember

A polynomial has terms where all powers are whole numbers like 0, 1, 2, 3, not fractions.

Degree means the biggest power of the variable in the polynomial.

You can add, subtract, and multiply polynomials just like you do with normal algebra.

Degree Of A Polynomial

Write the greatest index of the variable in the following polynomials.

Ex. (1) The greatest index of variable in the polynomial \(3x^2 + 4x\) is 2

Ex. (2) The greatest index of a variable in the polynomial \(7x^3 + 5x + 4x^5 + 2x^2\) is 5

The greatest index of the variable in the given polynomial is called the degree of the polynomial.

Teacher's Note

The degree tells us how big the polynomial is. If you have \(x^5\), the degree is 5 because 5 is the biggest power. It's like finding the tallest person in your class.

Exam Trick

Quick tip: Look at all the powers of x in the polynomial. The biggest number is your degree. Don't get confused by the order they are written in - just find the highest power.

Points to Remember

Degree is found by looking at the highest power of the variable.

The power can be 0, 1, 2, 3, 4, and so on for a polynomial.

A polynomial like \(5x + 3\) has degree 1 because the highest power is 1.

To Divide A Polynomial By A Monomial

Study the following examples and learn the method of division of polynomial by a monomial.

Ex. (1) \((6x^3 + 8x^2) \div 2x\)

Solution:

Explanation:

(i) \(2x \times 3x^2 = 6x^3\)

(ii) \(2x \times 4x = 8x^2\)

Quotient = \(3x^2 + 4x\)

Remainder = 0

Teacher's Note

Division of polynomials is like sharing - if you have 6x³ + 8x² and want to divide by 2x, you ask "what do I multiply 2x by to get each term?" This is the same as division in regular math.

Exam Trick

Remember: Division is the opposite of multiplication. So if you multiply 2x by 3x², you get 6x³. This helps you find what to write in the quotient.

Points to Remember

Division of polynomials works just like long division that you learned for numbers.

Keep dividing until the remainder is zero or is smaller than the divisor.

Always write your answer as quotient and remainder.

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