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Regard $\mathbb{R}[X,Y]$ as the polynomial ring $(\mathbb{R}[X])[Y]$. The coefficient ring $\mathbb{R}[X]$ is a Euclidean domain, hence a principal ideal domain. The element $X$ is a prime element: the quotient ring $\mathbb{R}[X]/(X)\cong \mathbb{R}$ is an integral domain.

The monic polynomial $Y^2+X$ has its other coefficients divisible by $X$, while $X^2\nmid X$. Part (a)(i), with $X$ playing the role of $p$ and $Y$ the indeterminate, proves that $Y^2+X$ is an irreducible polynomial.

Now use the automorphism of a polynomial ringIts inverse fixes $X$ and sends $Y$ to $Y-1$. A ring isomorphism preserves units and factorizations, hence preserves irreducible elements. Sincewe obtainApplying the Eisenstein criterion directly to the second expression with the prime element $X$ would fail, since $X\nmid 2$; the change of indeterminate supplies the required argument.

A polynomial is a finite sum of monomials with coefficients in a ring. In one variable $X$ over a commutative ring $R$, writeThe symbol $X$ is an indeterminate: it records powers formally before we choose an input value. Finite expressions are easy to add, multiply, evaluate, and differentiate, making polynomials a common language for exact algebra and approximate numerical analysis.

Consider the real polynomialThe expanded form displays the coefficients and the degree of a polynomial; the factored form displays its roots of a polynomial. At $X=3$, either description gives $p(3)=4$. At $X=-1$ and $X=2$, the factored form makes the zero values immediate. Keeping several descriptions of one object is useful because each makes a different question easy.

The graph shows another connection. The factor $X+1$ changes sign across $-1$, while $(X-2{)}^2$ stays nonnegative across $2$. Thus the graph crosses the axis at the simple root of a polynomial and touches it at the double root. Multiplicity of a root turns an algebraic repetition into visible local behavior.

For calculations, Horner's method evaluates a polynomial through nested multiplication. For modeling, a Taylor polynomial describes local behavior near a point, while polynomial interpolation constructs a polynomial matching prescribed data. Their usefulness comes with assumptions: a local approximation and an exact identity are different claims, and changing the coefficient ring can change which polynomial roots exist.

Polynomial evaluation replaces the indeterminate of a polynomial by an input value. For a fixed scalar $a$, the rule $E_a:p\mapsto p(a)$ is a linear functional on a vector space of polynomials, because evaluating a linear combination gives the same linear combination of the values. For $p(t)=1+2t+3t^2$, evaluation at zero gives one and evaluation at two gives seventeen. Polynomial interpolation uses these measurements to reconstruct a polynomial subject to a polynomial degree bound.

A polynomial in one indeterminate; its coefficients lie in a specified ring. For example, $3x^2+1$ is univariate, whereas $x+y$ involves two indeterminates.