An expression of the form a\(_{0}\)x\(^{n}\) + a\(_{1}\)x\(^{n - 1}\) + a\(_{2}\)x\(^{n - 2}\) + a\(_{3}\)x\(^{n - 3}\) + ..... + a\(_{n}\) where a\(_{0}\), a\(_{1}\), a\(_{2}\), a\(_{3}\), ....., a\(_{n}\) are given numbers (real or complex), n is a non-negative integer and x is a variable is called a polynomial in x.
a\(_{0}\), a\(_{1}\), a\(_{2}\), a\(_{3}\), etc., are called the coefficients of x\(^{n}\), x\(^{n - 1}\), x\(^{n - 2}\), x\(^{n - 3}\), etc., respectively.
a\(_{0}\)x\(^{n}\), a\(_{1}\)x\(^{n - 1}\), a\(_{2}\)x\(^{n - 2}\), a\(_{3}\)x\(^{n - 3}\), ....., a\(_{n}\) are called the terms of the polynomial.
a\(_{n}\) is called the constant term. Clearly, it is also the coefficient of x\(^{0}\).
If a\(_{0}\) ≠ 0, the polynomial is said to be of degree n and the term a\(_{0}\)x\(^{n}\) is called the leading term.
The general form of a polynomial of degree 1 is a\(_{0}\)x + a\(_{1}\)where a\(_{0}\) ≠ 0.
The general form of a polynomial of degree 2 is a\(_{0}\)x\(^{2}\) + a\(_{1}\)x + a\(_{2}\) where a\(_{0}\) ≠ 0.
A non-zero constant a\(_{0}\) itself is said to be a polynomial of degree 0 while a polynomial all of whose coefficients are zero is said to be a zero polynomial and is denoted by 0 and no degree is assigned to it.
Since a polynomial is an expression containing the variable x, it is denoted by f(x), p(x) or g(x) etc.
The value of a polynomial f(x) for x = a where a is real
number or a complex number is denoted by f(a).
In particular, if the coefficients a\(_{0}\), a\(_{1}\), a\(_{2}\),
a\(_{3}\), .... of a polynomial f(x) be all real numbers, the polynomial f(x)
is said to be a real polynomial.
Examples of polynomial:
(i) 7x\(^{2}\) + 5x - 3 is a polynomial in x of degree 2 or a quadratic polynomial in x.
(ii) 4x\(^{3}\) + 9x\(^{2}\) - 4x + 2 is a polynomial in x
of degree 3 or a cubic polynomial in x.
(iii) 5 - 2x\(^{\frac{5}{3}}\) + 9x\(^{2}\) is an expression but not a polynomial, since it contains a term containing x\(^{\frac{5}{3}}\) , where \(\frac{5}{3}\) is not a non-negative integer.
● Factorization
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