CBSE Class 9 Mathematics Polynomials Notes Set B

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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