Today is our topic of discussion Polynomial of One Variable .
Polynomial of One Variable
Let, r be a variable. Then, (i) a, (ii) ax+b, (iii) ax²+bx+c, (iv) ax³+bx²+cx+d ; expressions are polynomials of the variable x; where a, b, c, d are fixed numbers constants. Generally, the terms of a polynomial of the variable x have the form cx, where C (independent of x) is a fixed number (which may be zero) and p is a non-negative integer. If p is zero, the term becomes C, and if C is zero, the term is suppressed in the polynomial. C is called the coefficient of ™ in the term cr” and p is called the degree of the term.
The largest of the degrees of the terms appearing in a polynomial is called the degree of the polynomial. The term having the largest degree is called the leading term and the coefficient of the term having the largest degree is called the leading coefficient; the term with degree 0, that is, the term independent of variable x, is called the constant term.
For example, 2×6 – 3×5 – x¹ + 2x – 5 is a polynomial of the variable x, its degree is 6, its leading term is 26 leading coefficient is 2 and constant is -5. If a 0, the aforesaid (i) degree of the polynomial is 0, (ii) degree of the polynomial is 1, (iii) degree of the polynomial is 2, and (iv) degree of polynomial is 3. Any non zero constant (a 0) is a 0 degree polynomial of the variable (consider a = arº).
The number 0 is considered a zero-polynomial and the degree of a zero-polynomial is considered undefined A polynomial of the variable x is usually arranged in descending order of the degrees of its terms (the leading term appearing first and the constant term appearing last). This arrangement of a polynomial is called its standard form.
For convenience of using, polynomials of the variable x are denoted by symbols like P(x), Q(x) et catera. For example, P(x) = 2x² +7x+5. Such symbol P(x) indicates that the values of a polynomial depend on the values of x. If a specific number a is substituted for x in the polynomial, the value of the polynomial for that value of x is denoted by P(a).
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