# Polynomials Class 10 Notes Maths: Chapter 2

## Polynomialsa

- An algebraic expression, in which variables do not occur in the denominator, powers of variables are whole numbers and numerical coefficients of various terms are real numbers, iscalled a polynomial.
- General representation of a polynomial: f(x) = a
_{n}x^{n}+ an – 1 x^{n – 1}+ a_{n – 2}xn– 2+…….+ a_{1}x + a_{0}where, n = positive integer and the constants a_{0}, a_{1}, a_{2}…are called as coefficients. Here f(x) is a polynomial invariable x. - We can do the following operations while solving polynomials

(i) Addition of polynomials

(ii) Subtraction of polynomials

(iii) Multiplication of polynomials

(iv) Division of Polynomials - Polynomials can be classified on the basis of number of terms as :

(i) Monomial : One term, for example : 2x, x^{2}, 2x^{3}

(ii) Binomial : Two terms, for example : 6a – 5, x^{2}+1

(iii) Trinomial : Three terms, for example : m^{2}– m – 1, x+y+z4 - The highest value of exponents is called degree of polynomial. For example : x
^{2}– 3 is a second degree polynomial. a + 11 is a first degree polynomial. x^{4}– 16 is a fourth degree polynomial. Also, a constant term (such as 7,22, – 9) is a zero degree polynomial. - A polynomial of degree 0 is called a constant polynomial
- If α and β are the zeroes of p(x) = ax
^{2}+ bx + c and a ≠ 0, then

(i) α + β = – b/a, and

(ii) α β = c/a - A quadratic polynomial whose zeroes are α and β is given byp(x) or f(x) = x
^{2}– (α + β)x + αβ. - If α, β and γ are the zeroes of polynomial p(x)=ax
^{3}+ bx^{2}+ cx + d and a ≠ 0, then

(i) α + β + γ = –b/a

(ii) αβ + βγ + γα =ca

(iii) αβγ = –d/a - A cubic polynomial whose zeroes are α, β and γ is given byp(x) or f(x) = x
^{3}– (α + β + γ)x^{2}+ (αβ + βγ + γα)x – αβγ - A real number ‘a’ is a zero of a polynomialf(x) if f(a) = 0
- A polynomial ofdegree n can have atmostn real roots.
- Geometrically, the zeroes of a polynomial f(x) are the x-coordinates of the points where the graphy= f(x) intersects X-axis.