CBSE Class 9 Mathematics Polynomials Notes Set A

CBSE Class 9 Concepts for Polynomials. Learning the important concepts is very important for every student to get better marks in examinations. The concepts should be clear which will help in faster learning. The attached concepts made as per NCERT and CBSE pattern will help the student to understand the chapter and score better marks in the examinations.

IX Math

Ch 2: Polynomials

Chapter Notes

Top Definitions

1. A polynomial p(x) in one variable x is an algebraic expression in x of the form
p(x) = a_nxⁿ + a_n-1 x^n-1 + a_n-2 x^n-2 + .......a₂x² + a₁x + a₀ , where

(i) a₀,a₁, a₂, a_n  ......a are constants

(ii) x₀ ,x₁ ,x₂ ......x_n are variables

(iii) a₀,a₁, a₂, a_n  ......aare respectively the coefficients of x₀ ,x₁ ,x₂ ......x_n .

(iv) Each of a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 ,........... a₂x²+a₁x,+a₀,with a_n≠0,  is called a term of a polynomial.

2. A leading term is the term of highest degree.

3. Degree of a polynomial is the degree of the leading term.

4. A polynomial with one term is called a monomial.

5. A polynomial with two terms is called a binomial.

6. A polynomial with three terms is called a trinomial.

7. A polynomial of degree 1 is called a linear polynomial. It is of the form ax+b. For example: x-2, 4y+89, 3x-z.

8. A polynomial of degree 2 is called a quadratic polynomial. It is of the form ax² + bx + c. where a, b, c are real numbers and a¹ 0 For example:  x² - 2x +5 etc.

9. A polynomial of degree 3 is called a cubic polynomial and has the general form ax³ + bx² + cx +d. For example:  x³ + 2x² - 2x +5 etc.

10. A bi-quadratic polynomial p(x) is a polynomial of degree 4 which can be reduced to quadratic polynomial in the variable z = x² by substitution.

11. The zero polynomial is a polynomial in which the coefficients of all the terms of the variable are zero.

12. Remainder theorem: Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x – a, then remainder is p(a).

13. Factor Theorem: If p(x) is a polynomial of degree n≥ 1and a is any real number then (x-a) is a factor of p(x), if p(a) =0.

14. Converse of Factor Theorem: If p(x) is a polynomial of degree n≥ 1and a is any real number then p(a) =0 if (x-a) is a factor of p(x).

15. An algebraic identity is an algebraic equation which is true for all values of the variables occurring in it.

Top Concepts

1. The degree of non-zero constant polynomial is zero.

2. A real number ‘a’ is a zero/ root of a polynomial p(x) if p (a) = 0.

3. The number of real zeroes of a polynomial is less than or equal to the degree of polynomial.

4. Degree of zero polynomial is not defined.

5. A non zero constant polynomial has no zero.

6. Every real number is a zero of a zero polynomial.

7. Division algorithm: If p(x) and g(x) are the two polynomials such that degree of p(x) ³ degree of g(x) and g(x)≠ 0, then we can find polynomials q(x) and r(x) such that:

p (x) = g(x) q(x) + r(x)

where, r(x) =0 or degree of r(x) < degree of g(x).

8. If the polynomial p(x) is divided by (x+a), the remainder is given by the value of p (-a).

9. If the polynomial p(x) is divided by (x-a), the remainder is given by the value of p (a). 

10. If p (x) is divided by ax + b = 0; a ¹ 0, the remainder is given by

p (-b/a) ; a ≠ 0.

11. If p (x) is divided by ax - b = 0 , a ¹ 0 , the remainder is given by

p (-b/a) ; a ≠ 0.

12. A quadratic polynomial ax2 + bx+ c is factorised by splitting the middle term bx as px +qx so that pq =ac.

13. The quadratic polynomial ax2 + bx+ c will have real roots if and only if b2-4ac ≥ 0.

14. For applying factor theorem the divisor should be either a linear polynomial of the form x-a or it should be reducible to a linear polynomial.

Top Formulae

1. Quadratic identities:

a. (x+ y)² = x² + 2xy + y2

b. (x- y)² = x² - 2xy + y²

c. (x- y) (x + y) = x² - y²

d. (x+ a) (x + b) = x² + (a + b)x + ab

e. (x+ y + z)² = x² + y² + z² + +2xy + 2yz + 2zx

Here x, y, z are variables and a, b are constants

2. Cubic identities:

a. (x+ y)³ = x³ + y³ + 3xy(x + y)

b. (x - y)³ = x³ - y³ - 3xy(x - y)

c. x³ + y³ = (x + y)(x² - xy + y² )

d. x³ - y³ = (x - y)(x² + xy + y² )

e. x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx) 

f. If x +y + z = 0 then x³ + y³ + z³ = 3xyz

Here, x, y & z are variables.

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