A polynomial is an arithmetic expression consisting of variables and constants that involves four fundamental arithmetic operations and non-negative integer exponents of variables.
Polynomial in One Variable
An algebraic expression of the form
p(x) = an xn + an-1 xn - 1 + ........... + a2 x2 + a1 x + a0
Polynomial in one variable x of degree ‘n’ where a0, a1, a2, ...an are constants (an ≠ 0) and n is a whole number.
In general polynomials are denoted by f(x), g(x), p(t) , q(z) and r(x) and son on.
Note :
The coefficient of variables in the algebraic expression may have any real numbers, where as the powers of variables in polynomial must have only non-negative integral powers that is, only whole numbers. Recall that a0= 1 for all a.
Example :
Which of the following expressions are polynomials. If not give reason:
(i) (1/x2) + 3x - 4
Solution :
(1/x2) + 3x - 4 = x-2 + 3x - 4
Since it has negative exponent, the given expression is not a polynomial.
(ii) x2(x - 1)
Solution :
x2(x - 1) = x3 - x2
It is a polynomial.
(iii) (1/x)(x + 5)
Solution :
(1/x)(x + 5) = (x/x) + (5/x)
= 1 + 5 x-1
Since it has negative exponent, the given expression is not a polynomial.
(iv) (1/x-2) + (1/x-1) + 7
Solution :
= (1/x-2) + (1/x-1) + 7
Since it has negative exponent, the given expression is not a polynomial.
(v) √5 x2 + √3x + √2
Solution :
= √5 x2 + √3x + √2
The given expression is a polynomial.
(vi) m2 - ∛m + 7m - 10
Solution :
The given expression is having rational power, hence it is not a polynomial.
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