review-of-fundamentals

The parts of an expression connected by the signs + or – are called the terms of the expression. For example, \(x^{2}\) , \(2xy\) , and \(\sqrt{x}\) are the terms of the expression \(x^{2}+2xy+\sqrt{x}\) , and the expression \(2a-ab+c^{2}\) is made up of three terms: \(2a\) , \(-ab\) , and \(c^{2}\) .

A polynomial or more precisely a polynomial in \(x\) is an algebraic expression consisting of terms in the form \(ax^{k}\) where \(k\) is a nonnegative integers (that is zero or a natural number 0, 1, 2, …) and \(a\) is a real number called the coefficient of the term. For example, \[3,\quad x,\quad7x^{51},\quad-x^{3}+\frac{1}{\sqrt{14}}x-3,\quad\text{and }\quad7x^{5}-\pi x^{4}-\sqrt{2}x^{3}-x^{2}+x-3\] are all polynomials. In the last example, the coefficients are \(7,-\pi,-\sqrt{2},-1,1,\) and \(-3\) . Note that any constant is also a polynomial because it can be written as \(ax^{0}\) ; for example \(3=3x^{0}\) .

• Of course, instead of a polynomial in \(x\) , we may have polynomials in \(y\) or \(z\) or any other letter. For example, \(4z^{3}-0.5z+1\) is a polynomial in \(z\) .
• A polynomial that has only one term is called a monomial . For example, \(4x^{3}\) and \(-\sqrt{7}x^{2}\) are two monomials. A polynomial that has two terms is called a binomial , and a polynomial that has three terms is called a trinomial .
• The largest exponent in a polynomial is called the degree of the polynomial. In \(7x^{5}-\pi x^{4}-\sqrt{2}x^{3}-x^{2}+x-3\) , the largest power of \(x\) and hence the degree of the polynomial is 5.

Polynomials of degree 0, 1, 2, and 3 have special names. If \(a\neq0\) then

• When two or more terms differ only in their numerical coefficients, we say they are similar or like terms . For example, \(4x^{2}\) , \(\frac{3}{2}x^{2}\) , and \(-\sqrt{3}x^{2}\) are like terms but \(3x\) and \(3x^{2}\) are unlike terms.