Polynomials
- 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:
- \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
- \[{{\left( a\text{ }-\text{ }b \right)}^{2}}=\text{ }{{a}^{2}}-2ab+{{b}^{2}}\])
- \[~\left( a+b \right)\text{ }\left( a-b \right)\text{ }=\text{ }{{a}^{2}}-{{b}^{2}}\]
- \[{{\left( a\text{ }+\text{ }b\text{ }+\text{ }c \right)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ca\]
- \[{{(a+b)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab(a+b)={{a}^{3}}+{{b}^{3}}+3a{{b}^{2}}\]
- \[{{\left( a-b \right)}^{3}}=\text{ }{{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right)={{a}^{3}}-{{b}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}\]
- \[{{a}^{3}}+{{b}^{3}}=\left( {{a}^{2}}-ab+{{b}^{2}} \right)\]
- \[{{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\]
- \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=\left( a+b+c \right)\text{ }\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}-\text{ }ab\text{ }-\text{ }be\text{ }-\text{ }ca \right)\]
- lf a+ b+ c=0 then \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc.\]
- (x + a) (x + b) = x (a + b) x + ab
