#### Study Materials and Revision Notes for Ch 2 Polynomials Class 10th Maths

**Polynomials**

•

__An expression of the form p(x) = a__

*Polynomials:*_{0}+ a

_{1}x + a

_{2}x

^{2}+ .... + a

_{n}x

^{n}where a

_{n}≠ 0 is called a polynomial in variable x of degree n. where; a

_{0}, a

_{2 ....}a

_{n}are real numbers and each power of x is a non negative integer.Example: 3x

^{2 }+ 5x + 3 is a polynomial of degree two which is non negative integer.

√x + 5 is not a polynomial because degree of x is not a non negative integer.

• Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.

(ii) ax

(iii) ax

•

A polynomial which contains only constant term, is called a zero polynomial.

Example: 5, ax

•

Example: -3/2 is called zero of a polynomial p(x) = 2x + 3 because p(-3/2) = 2x + 3.

(i) A linear polynomial has at most one zero.

(ii) A Quadratic polynomial has at most two zeroes.

(iii) A Cubic polynomial has at most three zeroes.

(iv) A polynomial of degree n has at most n zeroes.

•

(i) Sum of zeroes = α + β = -b/a = (-coefficient of x)/(coefficient of x

(ii) Product of zeroes = α.β = c/a = (constant term)/(coefficient of x

(iii) A quadratic polynomial whose zeroes are α and β, is given by:

p(x) = k[x

•

(i) α + β + γ = -b/a = (-coefficient of x

(ii) αβ + βγ+ γα = c/a = (constant term of x)/(coefficient of x

(iii) α.β.γ = -d/a = (-constant term)/(coefficient of x

(iv) A cubic polynomial whose zeroes are α, β and γ, is given by:

p(x) = k[x

•

p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).

(ii) ax

^{2 }+ bx + c is a polynomial of degree 2 called quadratic polynomial.(iii) ax

^{3 }+ bx^{2 }+ cx + d is a polynomial of degree 3 cubic polynomial.•

__A polynomial of degree zero is called zero polynomial. Or,__*Zero Polynomial:*A polynomial which contains only constant term, is called a zero polynomial.

Example: 5, ax

^{0 }+ 3•

__A real number k is said to be zero of a polynomial p(x) if p(k) = 0.__*Zero of a polynomial:*Example: -3/2 is called zero of a polynomial p(x) = 2x + 3 because p(-3/2) = 2x + 3.

(i) A linear polynomial has at most one zero.

(ii) A Quadratic polynomial has at most two zeroes.

(iii) A Cubic polynomial has at most three zeroes.

(iv) A polynomial of degree n has at most n zeroes.

•

__If α,β are zeroes of polynomial p(x) = ax__*For quadratic polynomial:*^{2 }+ bx + c then:(i) Sum of zeroes = α + β = -b/a = (-coefficient of x)/(coefficient of x

^{2})(ii) Product of zeroes = α.β = c/a = (constant term)/(coefficient of x

^{2})(iii) A quadratic polynomial whose zeroes are α and β, is given by:

p(x) = k[x

^{2}- (α+β)x + αβ] where k is any real number.•

__If α,β and γ are zeroes of polynomial p(x) = ax__*For cubic polynomial:*^{3 }+ bx^{2 }+ cx + d then:(i) α + β + γ = -b/a = (-coefficient of x

^{2})/(coefficient of x^{3})(ii) αβ + βγ

^{3})(iii) α.β.γ = -d/a = (-constant term)/(coefficient of x

^{3})(iv) A cubic polynomial whose zeroes are α, β and γ, is given by:

p(x) = k[x

^{3}- (α+β+γ)x^{2}+ (αβ+βγ+γα)x - αβγ] where k is any real number.•

__: If p(x) and g(x) are any two polynomials where g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that:__*Division Algorithm*p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).