Notes of Ch 2 Polynomials| Class 10th Math

Revision Notes of Ch 2 Polynomials Class 10th Math

Topics in the Chapter

  • Polynomials
  • Zero Polynomial
  • Zero of a polynomial
  • Division algorithm


Polynomials

Polynomials: An expression of the form p(x) = a0 + a1x + a2x2 + .... + anxn where an ≠ 0 is called a polynomial in variable x of degree n. where; a0, a2 .... an are real numbers and each power of x is a non negative integer.Example: 3x+ 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.
(i) ax + b is polynomial of degree 1 called linear polynomial.
(ii) ax+ bx + c is  a polynomial of degree 2 called quadratic polynomial.
(iii) ax3 + bx+ cx + d is  a polynomial of degree 3 cubic polynomial.

Zero Polynomial: A polynomial of degree zero is called zero polynomial. Or,
A polynomial which contains only constant term, is called a zero polynomial.
Example: 5,  ax0 + 3

Zero of a polynomial: A real number k is said to be zero of a polynomial p(x) if p(k) = 0.
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.

For quadratic polynomial: If α,β are zeroes of polynomial p(x) = ax+ bx + c then:
(i) Sum of zeroes = α + β = -b/a = (-coefficient of x)/(coefficient of x2)
(ii) Product of zeroes = α.β = c/a = (constant term)/(coefficient of x2)
(iii) A quadratic polynomial whose zeroes are α and β, is given by:
p(x) = k[x2 - (α+β)x + αβ] where k is any real number.

For cubic polynomial: If α,β and γ are zeroes of polynomial p(x) = ax3 + bx+ cx + d then:
(i) α + β + γ = -b/a = (-coefficient of x2)/(coefficient of x3)
(ii) αβ + βγ + γα = c/a = (constant term of x)/(coefficient of x3)
(iii) α.β.γ = -d/a = (-constant term)/(coefficient of x3)
(iv) A cubic polynomial whose zeroes are α, β and γ, is given by:
p(x) = k[x3 - (α+β+γ)x2 + (αβ+βγ+γα)x - αβγ] where k is any real number.

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

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