These comprehensive RBSE Class 9 Maths Notes Chapter 2 Polynomials will give a brief overview of all the concepts.

## RBSE Class 9 Maths Chapter 2 Notes Polynomials

1. Unknown quantities in mathematical calculations are represented by symbols x, y, z, s, t The symbols a, b, c, etc. are used to denote constants. The branch of mathematics in which symbols, usually letters of the alphabets, representing numbers are combined according to the rules of arithmetic is known as algebra.

2. An algebraic expression is an expression that can contain ordinary numbers, variables (like x or y) and operators (like add, subtract, multiply and divide).

For example- 2x + 3y + 15, 3x + 9, x ÷\(\frac{1}{x}\), x + y, y^{3} + z etc. are algebraic expressions.

3. If an algebraic expression contains only one variable x and exponents of variable x are whole numbers then such an expression is called a polynomial in one variable which is x and is denoted by the symbol f(x), p(x), q(x) A polynomial in one variable x of degree n is an expression of the form-

p(x) = a_{n}x^{n} + a_{n-1} x^{n-1} + …… + a_{2}x^{2} + a_{1}x + a_{0}

a_{0}, a_{1}, a_{2},…., a_{n} are constants and a_{n} ≠ 0.

Also, a_{0}, a_{1,}a_{2},…., a_{n} are coefficients of x^{0}, x, x^{2},…., x^{n} respectively.

a_{n}x^{n}, a_{n-1}^{n-1},…., a_{0} are known as terms of the polymial P(x). Also, ‘n’ is called degree of the polynomial, i.e., the highest power of variable in a polynomial is called the degree of polynomial.

4. Polynomials having only one term are called monomial.

5. Polynomials having two terms are called binomials.

6. Polynomials having three terms are called trinomials.

7. Polynomials having degree 1 are called linear polynomials.

8. Polynomials having degree 2 are called quadratic polynomials.

9. Polynomials having degree 3 are called cubic polynomials.

10. The constant polynomial 0 or p(x) = 0 is called the zero polynomial. The degree of a zero polynomial is not defined.

11. Real number ‘a’ is a zero (or root) of the polynomial p(x) if p(a) = 0. In this case, a is also called a root of the equation p(x) = 0.

12. Zeros of polynomial: Some results-

- A zero of a polynomial need not be 0.
- . 0 may be a zero of a polynomial.
- Every linear polynomial has one and only one zero.
- A polynomial can have more than one zero.

13. A linear polynomial in one variable has a unique root. A non-zero constant polynomial has no root. Every real number is a root of the zero polynomial.

14. Standard form of polynomial: The standard form of a polynomial is that form in which terms are written in the decreasing order of their power, e., write each term in order of degree, from highest to lowest, left to right.

For example- x^{6} – ax^{5} + x^{4} – ax^{3} + 3x – a + 2 is in standard form.

15. When two or more polynomials are written as product of two or more quantities, then each expression of the product is known as its factor. This process of getting factors is known as factorization.

16. Factorization of quadratic trinomial expressions : A polynomial of degree two and having three terms is known as a quadratic trinomial expression. Its standard form is ax^{2} + bx + c, where a, b, c are constants and a ≠ 0.

17. To factorize ax^{2} + bx + c, we write split the middle term bx as two terms such that when added, they give bx but, when multiplied they give the product of the first and the third term (i.e., product of ax^{2} and c).

18. 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 the remainder is p(a).

19. Factor theorem: If p(x) is a polynomial of degree n ≥ 1 and a is any real number, then

(i) x – a is a factor of p(x), if p(a) = 0 and

(ii) p(a) = 0, ifx – a is a factor of p(x).

20. Algebraic identities : An identity is an equality which is true for all values of the variables. Some important identities are :

(i) (x+y)^{2}=x^{2}+2xy+9

(ii) (x – y)^{2} = x^{2} – zxy + y^{2}

(iii) x^{2}-y^{2}=(x + y)(x – y) .

(iv) (x+a)(x+b) = x^{2}+(a+b)x + ab

(v) (x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2xy+2yz+2zx

(vi) (x+y)^{3}=x^{3}+ y^{3}+2xy(x+y)

(vii) (x – y)^{3} = x^{3} – y^{3} – 3xy (x – y) = x^{3} -y^{3} – 3x^{2}y + 3xy^{2}

(viii) x^{3} +y^{3} + z^{3} – 3xyz = (x +y + z) (x^{2} +y^{2} + z^{2} – xy – yz – zx)

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