POLYNOMIALS IN ONE VARIABLE

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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 x_n + a_n-1 x^{n - 1} + ........... + a₂ x²+ a₁ x + a₀

Polynomial in one variable x of degree ‘n’ where a₀, a₁, a₂, ...a_n are constants (a_n ≠ 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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POLYNOMIALS IN ONE VARIABLE

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