- Polynomials in one Variable
- Zeroes of a Polynomial
- Remainder Theorem
- Factorisation of Polynomials
- Algebraic Identities
- Constants: A symbol having a fixed numerical value is called a constant.
- Variables: A symbol which may be assigned different numerical values is known as variable.
- Algebraic expressions: A combination of constants and variables. Connected by some or all of the operations +, -, X and is known as algebraic expression.
- Terms: The several parts of an algebraic expression separated by ‘+’ or ‘-‘ operations are called the terms of the expression.
- Polynomials: An algebraic expression in which the variables involved have only nonnegative integral powers is called a polynomial.
- 5x^{2} – 4x^{2} – 6x – 3 is a polynomial in variable x.
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- (ii) 5 + 8x^{2} + 4x^{-2} is an expression but not a polynomial.
Polynomials are denoted by p(x), q(x) and r(x)etc.
- Coefficients: In the polynomial x^{3} + 3x^{2} + 3x +1, coefficient of x^{3}, x^{2}, x are1,3,3 respectively
and we also say that +1 is the constant term in it.
- Degree of a polynomial in one variable: In case of a polynomial in one variable the highest
power of the variable is called the degree of the polynomial.
- Classification of polynomials on the basis of degree.
Degree Polynomial Example
- 1 Linear x +1, 2x + 3etc.
- 2 Quadratic ax^{2} + bx + c etc.
- 3 Cubic x^{3} + 3x^{2} +1 etc. etc.
- 4 Biquadratic x^{4} -1
Classification of polynomials on the basis of no. of terms
Polynomial & Examples. Monomial – S_{3}Sx_{3}^{1}Yetc.
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No. of terms
(i) 1
Binomial – (3 + 6x), (x – 5y) etc.
(ii) 2
(iii) 3
Trinomial- 2x^{2} + 4x + 2 etc. etc.
- Constant polynomial: A polynomial containing one term only, consisting a constant term is called a constant polynomial the degree of non-zero constant polynomial is zero.
- Zero polynomial: A polynomial consisting of one term, namely zero only is called a zero polynomial. The degree of zero polynomial is not defined.
- Zeroes of a polynomial: Let p(x) be a polynomial. If p(a) =0, then we say that is a zero of the polynomial of p(x).
- Remark: Finding the zeroes of polynomial p(x) means solving the equation p(x)=0.
- Remainder theorem: Let f (x) be a polynomial of degree n > I and let a be any real
number. When f(x) is divided by (x – a) then the remainder is f (a)
- Factor theorem: Let f(x) be a polynomial of degree n > 1 and let a be any real number.
- If f (a) = 0 then (x – a) is factor of f (x)
- If (x – a) is factor of f (x)then f (a) = 0
- Factor: A polynomial p(x) is called factor of q(x) divides q(x) exactly.
- Factorization: To express a given polynomial as the product of polynomials each of degree less than that of the given polynomial such that no such a factor has a factor of lower degree, is called factorization.