Maharashtra Board Class 9 Maths Part I Chapter 3 Polynomials PDF Download

Polynomials

Let's discuss.

p³ - \(\frac{1}{2}\)p² + p ; m² + 2n³ - \(\sqrt{3}\)m⁵; 6 are all algebraic expressions.

Teacher: Dear Students, consider each term of the expressions p³ - \(\frac{1}{2}\)p² + p, m² + 2n³ - \(\sqrt{3}\)m⁵, 6 and state the power of each variable.

Madhuri: In the expressions p³ - \(\frac{1}{2}\)p² + p powers of p are 3, 2, 1 respectively.

Vivek: Sir, in the expression m² + 2n³ - \(\sqrt{3}\)m⁵ the powers of the variable are 2, 3, 5 respectively.

Rahul: Sir, apparently there is no variable in the expression 6. But 6 = 6 × 1 = 6 × x⁰. Therefore, the power of the variable is 0.

Teacher: In all algebraic expressions given above the powers of the variable are positive integers or zero. i.e. whole numbers.

In an algebraic expression, if the powers of the variables are whole numbers then that algebraic expression is known as polynomial. 6 is also a polynomial.

6, - 7, \(\frac{1}{2}\), 0, \(\sqrt{3}\) etc. are constant numbers can be called as Constant polynomial. 0 is also a constant polynomial.

Are \(\sqrt{y}\) + 5 and \(\frac{1}{y}\) - 3 polynomials?

Sara: Sir, \(\sqrt{y}\) + 5 is not a polynomial, because \(\sqrt{y}\) + 5 = y^{\frac{1}{2}} + 5, here power of y is \(\frac{1}{2}\) which is not a whole number.

John: Sir, \(\frac{1}{y}\) - 3 is also not a polynomial because \(\frac{1}{y}\) - 3 = y⁻¹ - 3, here power of y is - 1 which is not a whole number.

Teacher: Write any five algebraic expressions which are not polynomials. Explain why these expressions are not polynomials? Justify your answer.

Is every algebraic expression a polynomial?

Is every polynomial an algebraic expression?

Teacher's Note

A polynomial has whole number powers only. Like how your school marks must be whole numbers from 0 to 100, not fractions.

Exam Trick

If you see a fraction or square root with the variable (like \(\frac{1}{x}\) or \(\sqrt{x}\)), it is NOT a polynomial. Remember: whole powers only!

Points to Remember

A polynomial has variables with whole number powers only.

Numbers like 5, -3, and 0 are also polynomials.

If a variable has a fraction or negative power, it is not a polynomial.

Polynomials

Introduction to Polynomials

Polynomials are mathematical expressions. They have variables and numbers. The variables have whole number powers only.

Examples of Polynomials

2x, 5x⁴ + x, m² - 3m, \(\frac{1}{2}\)y² - 2y + 5, x³ - 3x² + 5x

Polynomials are written as p(x), q(m), r(y) according to the variable used.

For example, p(x) = x³ + 2x² + 5x - 3, q(m) = m² + \(\frac{1}{2}\)m - 7, r(y) = y² + 5

Degree of a Polynomial in One Variable

Teacher: In the polynomial 2x⁷ - 5x + 9 which is the highest power of the variable?

Jija: Sir, the highest power is 7.

Teacher: In case of a polynomial in one variable, the highest power of the variable is called the Degree of the polynomial.

Now tell me, what is the degree of the given polynomial?

Ashok: Sir, the degree of the given polynomial 2x⁷ - 5x + 9 is 7.

Teacher: What is the degree of the polynomial 10?

Radha: 10 = 10 × 1 = 10 × x⁰ therefore the degree of the polynomial 10 is 0.

Teacher: Just like 10, degree of any non zero constant polynomial is 0. Degree of zero polynomial is not defined.

Degree of a Polynomial in More Than One Variable

The highest sum of the powers of variables in each term of the polynomial is the degree of the polynomial.

Ex. 3m³n⁶ + 7m²n³ - mn is a polynomial in two variables m and n. Degree of the polynomial is 9. (as sum of the powers 3 + 6 = 9, 2 + 3 = 5, 1 + 1 = 2)

Types of Polynomials (Based on Number of Terms)

Teacher's Note

Monomial means one term, binomial means two terms, trinomial means three terms. Like how "bicycle" has two wheels and "tricycle" has three wheels.

Exam Trick

Count the number of terms. Mono = 1, Bi = 2, Tri = 3. If you see \(\frac{1}{2}\)y² - 2y + 5, that is 3 terms, so it is trinomial.

Points to Remember

Monomial has only one term.

Binomial has two terms added or subtracted.

Trinomial has three terms.

Degree is the highest power of the variable.

Types of Polynomial (Based on Degree)

Polynomial: a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0 is a polynomial in x with degree n

a_n, a_{n-1}, ..., a_2, a_1, a_0 are the coefficients and a_n ≠ 0

Standard Form, Coefficient Form and Index Form of a Polynomial

p(x) = x - 3x² + 5 + x⁴ is a polynomial in x, which can be written in descending powers of its variable as x⁴ - 3x² + x + 5. This is called the standard form of the polynomial.

But in this polynomial there is no term having power 3 of the variable we can write it as 0x³. It can be added to the polynomial and it can be rewritten as x⁴ + 0x³ - 3x² + x + 5.

This form of the polynomial is called Index form of the polynomial.

Teacher's Note

Standard form means we write highest power first. Like arranging students from tallest to shortest. Index form includes all missing powers with zero.

Exam Trick

Always write polynomial in descending order of powers. If a power is missing, write it with coefficient 0. Example: x⁴ + 0x³ - 3x² + x + 5.

Points to Remember

Standard form means highest power comes first.

Index form shows all powers including missing ones with zero coefficients.

Coefficient form is just the numbers in order.

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