Is y=x a polynomial?

Question

If yes, what is its degree, leading coefficient and constant? Assume I know nothing and explain the meaning of all the math words you use.

-Clark

Answer 1

Thanks Clark for your questions. Math is a different language and I am a fluent speaker. I will do my best to give you a thorough answer you can understand.

First, I will explain the structure of math sentences, which are called equations. Every equation has an = sign in it separating two parts. The left part is called the left hand side, abbreviated LHS and the right part is the right hand side (RHS). These parts are equivalent, which means they can be substituted for each other in other equations. These parts by themselves without the = sign are called expressions. 

Expressions may have one or more parts, called terms, which are added together (subtraction is the same as adding a negative term). For example, 2x + 1 is an expression with two terms: 2x is the first term in this expression and 1 is the second term. Each term may also have 1 or more parts, called factors, multiplied together (dividing by a factor is the same as multiplying by its inverse). The term 2x has two factors: 2 (a constant) and x (a variable).

A term may have an expression as a factor by enclosing the expression within parentheses. For example (2x+1)(x-1) is a single term expression with two factors: 2x+1 is the first factor and x-1 is the second factor. This one term expression can be “simplified” by multiplying the two factors together to get 2x²-x-1 as a three term expression.

Polynomials are expressions. A one term polynomial is called a monomial, a two term polynomial is called a binomial, a three term polynomial is called a trinomial. If the polynomial has more than 3 terms, there is no special name for it and it is simply called a polynomial. Each term in a polynomial usually has 2 factors: a constant and a variable with a positive integer exponent. If the constant factor, called the coefficient, for a term is 1, the term will only show the variable. 1 is an implicit factor of any term since 1 is the multiplicative identity. Similarly, if the exponent of the variable is 1, it is not written explicitly. A constant without a variable factor is also a term in a polynomial. If you don’t see a constant term in a polynomial expression, then it is = 0. 0 is an implicit term of any expression since 0 is the additive identity.  

The degree of a polynomial is the highest degree of the terms in the polynomial. The degree of a term in a polynomial is the value of the exponent for the variable factor in that term. in the polynomial expression 2x²-x-1 there are 3 terms. The first term has a degree of 2 since the exponent of the variable x is 2. The second term has a degree of 1 since the exponent of x is implicitly 1. The third term has a degree of 0 since there is no explicit x factor (This factor is implicitly x⁰ = 1).

Back to the question: Is y=x a polynomial? y=x is an equation. y (the LHS of this equation) is a dependent variable (x is the independent variable) and is sometimes written as y=f(x). It depends on the calculated value of the expression on the RHS of this equation. Is the RHS of this equation a polynomial? Yes. Its degree is 1 because x = x¹. Its leading coefficient is 1 because x = 1x. The constant for this polynomial is 0 because x = x+0. We could write this polynomial as f(x) = 1x¹+0.