9th Class Mathematics Polynomials


Category : 9th Class



  • An expression of the form \[p(\operatorname{x})=+{{a}_{n}}{{\operatorname{x}}^{n}}+{{a}_{n-1}}......+{{a}_{2}}{{\operatorname{x}}^{2}}+{{a}_{1}}{{\operatorname{x}}^{2}}+{{a}_{0'}}\,\operatorname{where}{{a}_{0}},{{a}_{1}},a{{ & }_{2}},......,\]are real numbers \['n'\]is a non-negative integer and \[{{a}_{n}}\ne 0\] is called a polynomial of degree.


  • Each of \[{{a}_{n}}{{\operatorname{x}}^{n}},{{a}_{n-1}},......{{a}_{2}},{{x}^{2}},{{a}_{1}}\operatorname{x}\,and\,{{a}_{n}}\ne 0\]and a with is called a term of the polynomial p(x).


            Note: The power of variable in a polynomial must be a whole number.               

  • An expression of the form\[\frac{p\left( \operatorname{x} \right)}{q\left( \operatorname{x} \right)}\] where p(x) and q(x) are polynomials and \[q(\operatorname{x})\ne 0\]is called a rational expression.


            Note: Every polynomial is a rational expression, but every rational expression need not be a polynomial.


  • A polynomial d(x) is called a divisor of a polynomial p(x) if p(x) = d(x).q(x) for some polynomial q(x).


  • Polynomials of one term, two terms and three terms are called monomial, binomial and trinomial respectively.
  • A polynomial of degree one is called a linear polynomial.
  • A polynomial of degree two is called a quadratic polynomial.
  • A polynomial of degree three is called a cubic polynomial.
  • A polynomial of degree four is called a biquadratic polynomial.
  • A real number 'a' is a zero of a polynomial p(x) if p (a) = 0. 'a' is also called the root of the equation p(x) = 0.
  • Every linear polynomial in one variable has a unique zero.
  • A non-zero constant polynomial has no zero.
  • Every real number is a zero of the zero polynomial.
  • The degree of a non-zero constant polynomial is zero.
  • The degree of a zero polynomial is not defined.
  • If p(x) and g(x) are two polynomials such that degree of p(x) \[\ge \] degree of g(x) and g(x)\[\ne \]0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) q(x) + r(x).


  • Factor theorem:
  • Let f(x) be a polynomial of degree in > 1 and 'a' be any real number. Then

            (x - a) is a factor of f(x) if (a) = 0.

            (a) = 0 if (x - a) is a factor of f(x).

            If x - 1 is a factor of a polynomial of degree 'n' then the sum of its coefficients is zero.


  • Remainder theorem:
  • If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the line polynomial x - a (where 'a' is any real number) then the remainder is p (a).
  • We can express p(x) as p(x) = (x-a) q(x) +r(x) where q(x) is the quotient and r(x) is U remainder.
  • The process of writing an algebraic expression as the product of two or more algebra expressions is called factorization.


  • Some important identities:
  1. \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
  2. \[{{\left( a\text{ }-\text{ }b \right)}^{2}}=\text{ }{{a}^{2}}-2ab+{{b}^{2}}\])
  3. \[~\left( a+b \right)\text{ }\left( a-b \right)\text{ }=\text{ }{{a}^{2}}-{{b}^{2}}\]
  4. \[{{\left( a\text{ }+\text{ }b\text{ }+\text{ }c \right)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ca\]
  5. \[{{(a+b)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab(a+b)={{a}^{3}}+{{b}^{3}}+3a{{b}^{2}}\]
  6. \[{{\left( a-b \right)}^{3}}=\text{ }{{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right)={{a}^{3}}-{{b}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}\]
  7. \[{{a}^{3}}+{{b}^{3}}=\left( {{a}^{2}}-ab+{{b}^{2}} \right)\]
  8. \[{{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\]
  9. \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=\left( a+b+c \right)\text{ }\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}-\text{ }ab\text{ }-\text{ }be\text{ }-\text{ }ca \right)\]
  10. lf a+ b+ c=0 then \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc.\]
  11. (x + a) (x + b) = x (a + b) x + ab


Other Topics

Notes - Polynomials

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