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
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Feb 22, 24 04:21 PM
Feb 22, 24 04:15 PM
Feb 22, 24 02:30 PM
Feb 19, 24 11:57 PM
Feb 19, 24 11:14 PM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.