Monomials and Polynomials: Definitions, Properties, and Operations

Definition of a Monomial

A monomial in the variables \(x_1,x_2,\dots,x_n\) is an algebraic expression consisting of a numerical coefficient multiplied, where present, by powers of the variables with non-negative integer exponents.

Once reduced to standard form, a monomial may be written as

\[ a x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n}, \]

where \(a\) is a real number, called the coefficient, the \(x_1,x_2,\dots,x_n\) are the variables, and the exponents \(\alpha_1,\alpha_2,\dots,\alpha_n\) are non-negative integers, that is, elements of the set

\[ \mathbb{N}_0=\{0,1,2,3,\dots\}. \]

The following are monomials, for instance:

\[ 5x^2y^3,\qquad -\frac{3}{4}ab^2,\qquad x^4,\qquad -7. \]

In the monomial \(x^4\) the coefficient \(1\) is left unwritten; likewise, in the monomial \(-x^4\) the coefficient is \(-1\).

A monomial need not be presented in standard form at the outset. Consider, for example,

\[ 2x\cdot 3xy^2. \]

Multiplying the numerical factors and collecting the powers of the same variable, we obtain

\[ 2x\cdot 3xy^2=6x^2y^2. \]

The given expression is therefore a monomial, since it is built exclusively from products of numerical factors and powers of variables with non-negative integer exponents.

Every real number is likewise a monomial. A non-zero real number is called a constant monomial, since it contains no variables, whereas the number \(0\) is called the zero monomial.

The condition imposed on the exponents is essential. The following, for instance, are not monomials:

\[ \frac{1}{x},\qquad x^{-2},\qquad \sqrt{x},\qquad x^{\frac{3}{2}},\qquad 2^x. \]

Indeed, in the first two expressions the variable occurs with a negative exponent; in the next two it occurs with a non-integer exponent; in the last it appears as an exponent rather than as the base of a power.

In particular, in a monomial a variable may not occur in a denominator, under a radical sign, or in an exponent. A purely numerical denominator, on the other hand, is admissible, since it can be absorbed into the coefficient. For example,

\[ \frac{x^2y}{3}=\frac{1}{3}x^2y \]

is a monomial.

Not every algebraic expression is a monomial. The expression

\[ 3x^2y+1 \]

is a sum of two monomials that are not like, and hence cannot be collected into a single monomial. It is instead a polynomial, as we shall see in the sections that follow.

Standard Form, Coefficient and Variable Part

A monomial is said to be written in standard form when all its numerical factors have been multiplied together and each variable occurs exactly once, raised to its own exponent.

To obtain the standard form one uses the commutative and associative laws of multiplication, together with the rule for the product of powers with the same base:

\[ x^m x^n=x^{m+n}. \]

Consider the monomial

\[ 2x^2\cdot(-3)y\cdot x^4y^2. \]

Multiplying the numerical factors and adding the exponents of the powers with the same base, we obtain

\[ 2\cdot(-3)\cdot x^{2+4}y^{1+2}=-6x^6y^3. \]

The standard form of the given monomial is accordingly

\[ -6x^6y^3. \]

The order of the variable factors does not alter the monomial, since multiplication is commutative. The expressions

\[ 3x^2y,\qquad 3yx^2,\qquad 3xyx \]

therefore represent one and the same monomial. The first two are already in standard form, whereas the third is not, because the variable \(x\) occurs in two distinct factors. For the sake of uniform notation, the variables are usually arranged in alphabetical order.

In a non-zero monomial written in standard form, the numerical factor, sign included, is called the coefficient; the product of the powers of the variables constitutes the variable part.

In the monomial

\[ -6x^6y^3 \]

the coefficient is \(-6\), while the variable part is

\[ x^6y^3. \]

When the coefficient is \(1\) it is omitted; when it is \(-1\), only the minus sign is written:

\[ x^2y=1\cdot x^2y, \qquad -x^2y=-1\cdot x^2y. \]

A non-zero constant monomial has the number itself as its coefficient and is regarded as having variable part \(1\). In the monomial

\[ -7, \]

for instance, the coefficient is \(-7\) and the variable part is \(1\).

Variables with exponent \(0\) are not displayed in the standard form, since they correspond to the multiplicative factor \(1\). The variable part therefore contains only those variables whose exponent is positive.

Consider, finally,

\[ -\frac{2}{3}x\cdot 9xy^2. \]

Reduction to standard form leads to

\[ -\frac{2}{3}\cdot 9\cdot x^2y^2=-6x^2y^2. \]

Once an order of the variables has been fixed, every non-zero monomial possesses a unique standard form. This uniqueness makes it possible to compare two monomials at a glance, and it will be fundamental in defining equal, like and opposite monomials.

The zero monomial is a special case. Indeed, any product whose coefficient is \(0\) represents the same zero expression:

\[ 0x^2y=0a^5b^3=0. \]

For this reason it is written simply as \(0\), and no determinate variable part is assigned to it.

Degree of a Monomial

The exponents occurring in the variable part of a monomial describe its degree. This notion measures how many times, in total or with respect to a particular variable, the variable factors enter into the product.

Consider the non-zero monomial

\[ -5x^2y^3z. \]

The degree with respect to a variable is the exponent with which that variable occurs in the standard form of the monomial. Thus the monomial above has degree \(2\) with respect to \(x\), degree \(3\) with respect to \(y\), and degree \(1\) with respect to \(z\).

If a variable does not occur in the variable part, its exponent is taken to be \(0\). The same monomial therefore has degree \(0\) with respect to the variable \(t\), say.

The total degree, or simply the degree, of a non-zero monomial is the sum of the exponents of all the variables occurring in its standard form.

In our example we have

\[ 2+3+1=6. \]

The monomial

\[ -5x^2y^3z \]

is thus a monomial of total degree \(6\).

The coefficient has no bearing on the degree. The monomials

\[ 4x^2y^3,\qquad -7x^2y^3,\qquad \frac{1}{2}x^2y^3 \]

all have total degree \(5\), since they share the same variable part.

The order in which the variables are written is likewise immaterial. For example,

\[ 3a^2bc^4 \]

has degree \(2\) with respect to \(a\), degree \(1\) with respect to \(b\), degree \(4\) with respect to \(c\), and total degree

\[ 2+1+4=7. \]

A non-zero constant monomial has variable part equal to \(1\) and contains no variable with positive exponent. Its degree is accordingly \(0\). For instance,

\[ \deg(-8)=0. \]

The zero monomial, by contrast, is a special case. Since it may be written formally with any variable part whatsoever,

\[ 0=0x=0x^2y^5=0a^{10}b, \]

no determinate degree can be assigned to it. For this reason, in elementary algebra the degree of the zero monomial is left undefined.

The distinction between the degree with respect to a variable and the total degree becomes especially important when several variables are present. In the monomial

\[ 2x^4y^2, \]

the degree with respect to \(x\) is \(4\), the degree with respect to \(y\) is \(2\), while the total degree is \(6\). These three pieces of information must not be confused.

Equal, Like and Opposite Monomials

In order to compare two monomials correctly, they must first be written in standard form. Only then can the coefficient and the variable part of each be read off at once.

Two non-zero monomials are said to be equal when, once reduced to standard form, they have the same coefficient and the same variable part.

Consider, for example,

\[ 2x^2\cdot 3y \qquad\text{and}\qquad 6yx^2. \]

Reducing both monomials to standard form, we obtain

\[ 2x^2\cdot 3y=6x^2y, \qquad 6yx^2=6x^2y. \]

The two monomials are therefore equal, even though they were written differently to begin with.

Two non-zero monomials are said to be like when they have the same variable part, irrespective of their coefficients.

The monomials

\[ 3x^2y,\qquad -7x^2y,\qquad \frac{1}{2}x^2y \]

are like, since they all have variable part

\[ x^2y. \]

The monomials

\[ 3x^2y \qquad\text{and}\qquad 3xy^2 \]

are not like, however, since the variables \(x\) and \(y\) occur with different exponents.

Two equal monomials are necessarily like, but two like monomials need not be equal. To be equal, indeed, they must agree in both variable part and coefficient.

Two non-zero monomials are said to be opposite when they have the same variable part and opposite coefficients.

For example,

\[ 5a^2b^3 \qquad\text{and}\qquad -5a^2b^3 \]

are opposite monomials. Their sum is the zero monomial:

\[ 5a^2b^3-5a^2b^3=0. \]

More generally, if

\[ M=ax_1^{\alpha_1}\cdots x_n^{\alpha_n}, \]

the monomial opposite to \(M\) is

\[ -M=-ax_1^{\alpha_1}\cdots x_n^{\alpha_n}. \]

Once again the zero monomial is a special case. It is equal only to itself, and it is also its own opposite, since

\[ -0=0. \]

No determinate variable part is assigned to it; for this reason, when speaking of like monomials one ordinarily restricts attention to non-zero monomials.

Likeness of monomials is fundamental in algebraic operations: within an algebraic sum, only like monomials may be collected directly into a single term. For example,

\[ 3x^2y-7x^2y=-4x^2y, \]

whereas in the expression

\[ 3x^2y-7xy^2 \]

the two terms remain distinct, since they are not like.

Operations on Monomials

Operations on monomials rest on the properties of the real numbers, of multiplication and of powers. Before carrying out any calculation it is advisable to reduce the monomials to standard form, so that the coefficient and the variable part may be read off at once.

Addition and Subtraction

In addition and subtraction, like monomials may be collected into a single monomial, since they share the same variable part.

In that case one adds the coefficients algebraically and leaves the variable part unchanged:

\[ ax_1^{\alpha_1}\cdots x_n^{\alpha_n} + bx_1^{\alpha_1}\cdots x_n^{\alpha_n} = (a+b)x_1^{\alpha_1}\cdots x_n^{\alpha_n}. \]

For example,

\[ 7x^2y-3x^2y=(7-3)x^2y=4x^2y. \]

Similarly,

\[ -5a^3b+2a^3b=(-5+2)a^3b=-3a^3b. \]

If the sum of the coefficients vanishes, the result is the zero monomial:

\[ 4x^2y-4x^2y=0. \]

When two non-zero monomials are not like, their sum is still a perfectly well-defined algebraic expression, but the two terms cannot be collected into a single monomial. For example,

\[ 3x^2y+5xy^2 \]

is not a monomial, since the variable parts \(x^2y\) and \(xy^2\) are different.

Multiplication

The product of two monomials is always a monomial. To compute it one multiplies the coefficients and adds the exponents of the powers with the same base.

If

\[ M=ax_1^{\alpha_1}\cdots x_n^{\alpha_n} \qquad\text{and}\qquad N=bx_1^{\beta_1}\cdots x_n^{\beta_n}, \]

then

\[ MN = abx_1^{\alpha_1+\beta_1}\cdots x_n^{\alpha_n+\beta_n}. \]

Consider, for example,

\[ (2x^2y)(-3xy^4). \]

Multiplying the coefficients and applying the rule for the product of powers with the same base, we obtain

\[ (2x^2y)(-3xy^4) = -6x^{2+1}y^{1+4} = -6x^3y^5. \]

If the two monomials are non-zero, the degree of the product equals the sum of their degrees:

\[ \deg(MN)=\deg(M)+\deg(N). \]

In the preceding example the two factors have degree \(3\) and degree \(5\) respectively, while the product has degree

\[ 3+5=8. \]

Powers

To raise a monomial to a positive integer exponent, one raises the coefficient to that exponent and multiplies each exponent of the variable part by the exponent of the power.

If

\[ M=ax_1^{\alpha_1}\cdots x_n^{\alpha_n} \]

and \(m\) is a positive integer, then

\[ M^m = a^m x_1^{m\alpha_1}\cdots x_n^{m\alpha_n}. \]

For example,

\[ \left(-2x^3y^2\right)^3 = (-2)^3x^{3\cdot3}y^{2\cdot3} = -8x^9y^6. \]

The sign of the result depends on the sign of the coefficient and on the parity of the exponent. If the coefficient is non-zero, a power with even exponent has a positive coefficient, whereas a power with odd exponent retains the sign of the original coefficient. If the monomial is the zero monomial, each of its powers with positive integer exponent is again equal to \(0\).

If the monomial is non-zero, the degree of its power equals the product of the degree of the monomial and the exponent:

\[ \deg(M^m)=m\deg(M). \]

For every non-zero monomial one sets, in addition,

\[ M^0=1. \]

The case \(0^0\), on the other hand, is left undefined.

Division

Division of monomials does not always yield a monomial. Given two non-zero monomials

\[ M=ax_1^{\alpha_1}\cdots x_n^{\alpha_n} \]

and

\[ N=bx_1^{\beta_1}\cdots x_n^{\beta_n}, \qquad b\neq0, \]

one says that \(M\) is divisible by \(N\) when there exists a monomial \(Q\) such that

\[ M=NQ. \]

For such a monomial to exist, the exponent of each variable in the dividend must be greater than or equal to the corresponding exponent in the divisor. Hence one must have

\[ \alpha_1\geq\beta_1,\qquad \alpha_2\geq\beta_2,\qquad \dots,\qquad \alpha_n\geq\beta_n. \]

When these conditions are satisfied, the quotient monomial is

\[ Q=\frac{a}{b}x_1^{\alpha_1-\beta_1}\cdots x_n^{\alpha_n-\beta_n}. \]

For example, the monomial \(12x^5y^3\) is divisible by \(3x^2y\), the quotient monomial being

\[ 4x^{5-2}y^{3-1}=4x^3y^2, \]

and indeed

\[ (3x^2y)(4x^3y^2)=12x^5y^3. \]

The monomial \(6x^2y\), by contrast, is not divisible by \(3x^4\), since computing the exponents would require

\[ x^{2-4}=x^{-2}, \]

which cannot occur in the variable part of a monomial.

The zero monomial is divisible by every non-zero monomial, the quotient being the zero monomial. Division by the zero monomial is never defined.

If \(M\) and \(N\) are non-zero monomials and \(M\) is divisible by \(N\), the degree of the quotient is given by the difference of their degrees:

\[ \deg(Q)=\deg(M)-\deg(N). \]

Addition, subtraction, multiplication, powers and division therefore behave in quite different ways. In addition and subtraction only like monomials may be collected directly; the product and every power with positive integer exponent are again monomials; the power with exponent \(0\) is defined only for non-zero monomials; finally, a non-zero monomial is divisible by another non-zero monomial only when every exponent of the dividend is greater than or equal to the corresponding exponent of the divisor.

Definition of a Polynomial

A polynomial in the variables \(x_1,x_2,\dots,x_n\) is a finite algebraic sum of monomials in those same variables.

In general, a polynomial may therefore be written as

\[ P(x_1,x_2,\dots,x_n)=M_1+M_2+\cdots+M_k, \]

where \(M_1,M_2,\dots,M_k\) are monomials and \(k\) is a positive integer.

The following are polynomials, for instance:

\[ 3x^2y-5xy^3+7, \qquad a^4-2a^2b+b^2, \qquad 4x^3-x+1. \]

Each of the expressions above is made up of finitely many monomials joined by the signs \(+\) and \(-\). Subtraction calls for no separate definition, since subtracting a monomial amounts to adding its opposite:

\[ 3x^2-5x=3x^2+(-5x). \]

Every non-zero monomial is thus also a polynomial consisting of a single term. In particular,

\[ 6x^3y^2\qquad\text{and}\qquad -4 \]

are polynomials. The zero monomial is a polynomial as well, called the zero polynomial; in its reduced form it possesses no non-zero term.

In the case of a single variable \(x\), a polynomial may be written in the form

\[ P(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n, \]

where \(a_0,a_1,\dots,a_n\) are real numbers and the exponents of \(x\) are non-negative integers.

The coefficients need not all be non-zero. For example,

\[ P(x)=2x^5-3x^2+1 \]

may also be written as

\[ P(x)=1+0x-3x^2+0x^3+0x^4+2x^5. \]

Terms with zero coefficient are normally omitted, since they make no contribution to the value of the expression.

The word finite is essential. An expression involving an infinite sum of powers, such as

\[ 1+x+x^2+x^3+\cdots, \]

is not a polynomial, since it has infinitely many terms.

The conditions already met for monomials continue to apply here. The following, for instance, are not polynomials:

\[ \frac{1}{x}+2, \qquad \sqrt{x}+x, \qquad x^{\frac{3}{2}}-1, \qquad 2^x+x. \]

In these expressions the variable occurs, respectively, with a negative exponent, under a radical sign, with a non-integer exponent, and as the exponent of a power.

It is worth observing that one and the same polynomial may be presented in seemingly different ways. For example,

\[ 2x^2+3x-x^2-5x+4 \]

is a polynomial, but it contains terms that may be collected. After the like monomials have been added, it becomes

\[ x^2-2x+4. \]

The two expressions represent the same polynomial. The second is written in reduced form, a notion that we shall examine in the next section.

Terms, Coefficients and Reduced Form

The monomials making up a polynomial are called the terms of the polynomial. Consider, for example,

\[ P(x,y)=4x^3y-2xy^2+5x-7. \]

Its terms are

\[ 4x^3y,\qquad -2xy^2,\qquad 5x,\qquad -7. \]

The sign of each term is included in its coefficient. In the polynomial above, therefore, the coefficients are, respectively,

\[ 4,\qquad -2,\qquad 5,\qquad -7. \]

The term containing no variables is called the constant term. In the example under consideration, the constant term is \(-7\). If a polynomial contains no constant term, its constant term is taken to be \(0\).

A polynomial may contain like terms, that is, monomials with the same variable part. Such terms may then be collected by adding their coefficients algebraically.

Consider the polynomial

\[ 3x^2y-5xy^2+4x^2y+2xy^2-7. \]

The terms \(3x^2y\) and \(4x^2y\) are like, as are the terms \(-5xy^2\) and \(2xy^2\). Collecting them, we obtain

\[ (3+4)x^2y+(-5+2)xy^2-7, \]

and hence

\[ 7x^2y-3xy^2-7. \]

A polynomial is said to be written in reduced form when all its terms are monomials in standard form and no two terms are like.

To reduce a polynomial thus means first to reduce each of its terms to standard form and then to collect the like terms. Terms whose coefficient turns out to be \(0\) are deleted.

For example,

\[ 2x\cdot 3xy-4x^2y+x^2y+5 \]

first becomes

\[ 6x^2y-4x^2y+x^2y+5, \]

and then

\[ (6-4+1)x^2y+5=3x^2y+5. \]

The reduced form makes the actual structure of a polynomial evident at once. Indeed, two seemingly different expressions represent the same polynomial precisely when, after reduction, they exhibit the same terms with the same coefficients.

For example,

\[ 2x^2+3x-x^2-5x+4 \]

and

\[ x^2-2x+4 \]

represent the same polynomial, since the first expression reduces to the second.

In the case of a polynomial in a single variable,

\[ P(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n, \]

the number \(a_k\) is the coefficient of \(x^k\). If \(a_k\neq0\), the monomial \(a_kx^k\) is a term of the polynomial and has degree \(k\); if instead \(a_k=0\), the corresponding term is omitted from the notation.

For example, in the polynomial

\[ P(x)=3x^4-2x+1, \]

the coefficients of \(x^3\) and \(x^2\) both vanish. The polynomial may in fact also be written in the form

\[ P(x)=3x^4+0x^3+0x^2-2x+1. \]

Terms with zero coefficient are normally omitted, but regarding them as implicitly present makes it easier to compare different polynomials and to carry out operations on them in an orderly fashion.

Degree and Classification of Polynomials

Degree of a Polynomial

The degree of a non-zero polynomial is the greatest total degree among the monomials occurring in its reduced form.

The polynomial must therefore be reduced before its degree is determined, since adding like terms may alter or eliminate the terms of highest degree.

Consider, for example,

\[ P(x)=x^3-x^3+2x+1. \]

The two third-degree terms cancel and the polynomial reduces to

\[ P(x)=2x+1. \]

Its degree is therefore \(1\), not \(3\).

In the case of a polynomial in a single variable written in the reduced form

\[ P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0, \qquad a_n\neq0, \]

the degree is the largest exponent with which the variable \(x\) occurs:

\[ \deg(P)=n. \]

The coefficient \(a_n\) is called the leading coefficient, while the term \(a_nx^n\) is called the leading term.

For example, in the polynomial

\[ P(x)=-3x^5+2x^2-x+4, \]

the degree is \(5\), the leading coefficient is \(-3\) and the leading term is \(-3x^5\).

A non-zero constant polynomial has degree \(0\). For instance,

\[ \deg(7)=0. \]

The zero polynomial, by contrast, is a special case. Since it may be written with terms of any degree whatsoever,

\[ 0=0x=0x^2=0x^{10}, \]

in elementary algebra its degree is left undefined.

For a polynomial in several variables one may also consider the degree with respect to a given variable, defined as the largest exponent with which that variable occurs in the non-zero terms of the reduced form.

Consider the polynomial

\[ P(x,y)=3x^2y-5xy^3+7. \]

Its three terms have total degrees \(3\), \(4\) and \(0\) respectively. The total degree of the polynomial is therefore

\[ \deg(P)=4. \]

The degree with respect to \(x\), on the other hand, is \(2\), while the degree with respect to \(y\) is \(3\).

Classification by Number of Terms

Once a non-zero polynomial has been reduced, it may be classified according to the number of its terms.

A polynomial consisting of a single non-zero term is a monomial; a polynomial consisting of two terms is a binomial; a polynomial consisting of three terms is a trinomial.

For example,

\[ 4x^3,\qquad x^2-1,\qquad x^2+3x+2 \]

are respectively a monomial, a binomial and a trinomial.

The zero polynomial is a special case: it coincides with the zero monomial, but in its reduced form it possesses no non-zero term and is not classified according to the number of terms.

This classification, too, must be carried out after reduction. The expression

\[ x^2+3x-2x+1 \]

initially contains four terms, but it reduces to

\[ x^2+x+1, \]

and is therefore a trinomial.

Ordered and Complete Polynomials

A polynomial in one variable is said to be ordered when its terms are arranged according to ascending or descending powers of the variable.

The polynomial

\[ 3x^4-2x^3+x-5 \]

is ordered by descending powers of \(x\), whereas

\[ -5+x-2x^3+3x^4 \]

is ordered by ascending powers.

A polynomial of degree \(n\) is said to be complete with respect to the variable \(x\) when it contains, with non-zero coefficients, every power of \(x\) from \(n\) down to \(0\).

For example,

\[ 2x^4-x^3+3x^2+x-5 \]

is a complete polynomial of the fourth degree, since it contains the powers

\[ x^4,\qquad x^3,\qquad x^2,\qquad x,\qquad x^0. \]

The polynomial

\[ 2x^4+3x-5 \]

is incomplete, on the other hand, since it contains no terms in \(x^3\) and in \(x^2\). The coefficients corresponding to those powers are implicitly equal to \(0\).

Homogeneous Polynomials

A non-zero polynomial in several variables is said to be homogeneous when all its terms have the same total degree.

For example,

\[ P(x,y)=3x^3-2x^2y+5xy^2-y^3 \]

is homogeneous of degree \(3\), since each term has total degree \(3\).

The polynomial

\[ Q(x,y)=x^2+xy+y \]

is not homogeneous: the first two terms have degree \(2\), while the term \(y\) has degree \(1\).

Monic Polynomials

A non-zero polynomial in one variable is said to be monic when its leading coefficient is equal to \(1\).

The polynomial

\[ x^4-3x^2+2x-1 \]

is monic, whereas

\[ 2x^4-3x^2+2x-1 \]

is not, since its leading coefficient is \(2\).

Degree, number of terms, ordering, completeness, homogeneity and monicity describe different aspects of the structure of a polynomial. One and the same polynomial may therefore belong to several categories at once: it may, for instance, be a trinomial, ordered, incomplete and monic.

Operations on Polynomials

The sum, the difference and the product of two polynomials are again polynomials. Consequently, every power of a polynomial with positive integer exponent is a polynomial as well. One says accordingly that the set of polynomials is closed under addition, subtraction and multiplication.

Before carrying out the calculations it is advisable to reduce each polynomial and, in the case of polynomials in a single variable, to arrange the terms according to ascending or descending powers of the variable.

Addition

To add two polynomials one collects the like terms, adding their coefficients algebraically.

Consider

\[ P(x)=3x^3-2x^2+5x-1 \]

and

\[ Q(x)=-x^3+4x^2-3x+6. \]

Arranging the like terms in the same order, we obtain

\[ \begin{aligned} P(x)+Q(x) &=(3x^3-2x^2+5x-1)+(-x^3+4x^2-3x+6)\\ &=(3-1)x^3+(-2+4)x^2+(5-3)x+(-1+6)\\ &=2x^3+2x^2+2x+5. \end{aligned} \]

The sum may also be carried out by writing the polynomials one beneath the other, taking care to place terms with the same variable part in the same column.

For two non-zero polynomials \(P\) and \(Q\), the degree of the sum satisfies

\[ \deg(P+Q)\leq\max\{\deg(P),\deg(Q)\}, \]

provided \(P+Q\) is not the zero polynomial.

The degree of the sum may be smaller than the larger of the degrees of the two summands when the terms of highest degree cancel. For example,

\[ (2x^3+x)+(-2x^3+4)=x+4. \]

Both polynomials have degree \(3\), whereas their sum has degree \(1\).

Subtraction

To subtract a polynomial means to add its opposite. The opposite of a polynomial is obtained by changing the sign of every one of its terms.

If

\[ Q(x)=x^2-3x+5, \]

then

\[ -Q(x)=-x^2+3x-5. \]

Consider now

\[ P(x)=4x^2+x-2. \]

The difference \(P(x)-Q(x)\) is

\[ \begin{aligned} P(x)-Q(x) &=(4x^2+x-2)-(x^2-3x+5)\\ &=4x^2+x-2-x^2+3x-5\\ &=3x^2+4x-7. \end{aligned} \]

The brackets are essential: a minus sign placed before a polynomial changes the sign of every term within it.

Multiplication

The product of two polynomials is computed by repeated application of the distributive law: every term of the first polynomial must be multiplied by every term of the second.

Consider

\[ P(x)=2x-3 \qquad\text{and}\qquad Q(x)=x^2+4x+1. \]

We have

\[ \begin{aligned} P(x)Q(x) &=(2x-3)(x^2+4x+1)\\ &=2x(x^2+4x+1)-3(x^2+4x+1)\\ &=2x^3+8x^2+2x-3x^2-12x-3\\ &=2x^3+5x^2-10x-3. \end{aligned} \]

Once the distributive law has been applied, the result must be reduced by collecting any like terms.

The product of two non-zero polynomials is never the zero polynomial. Moreover, its degree equals the sum of the degrees of the factors:

\[ \deg(PQ)=\deg(P)+\deg(Q). \]

In the preceding example,

\[ \deg(P)=1, \qquad \deg(Q)=2, \]

and indeed

\[ \deg(PQ)=1+2=3. \]

This property follows from the fact that the leading term of the product is obtained by multiplying the leading terms of the two factors. If

\[ P(x)=a_nx^n+\cdots \qquad\text{and}\qquad Q(x)=b_mx^m+\cdots, \]

with \(a_n\neq0\) and \(b_m\neq0\), the leading term of the product is

\[ a_nb_mx^{n+m}. \]

Since the product \(a_nb_m\) of two non-zero real numbers is again non-zero, the degree of the product is \(n+m\).

Powers

To raise a polynomial to a positive integer exponent means to multiply it by itself as many times as the exponent indicates.

For example,

\[ P(x)^3=P(x)\cdot P(x)\cdot P(x). \]

Consider the polynomial

\[ P(x)=x+2. \]

Its second power is

\[ \begin{aligned} P(x)^2 &=(x+2)(x+2)\\ &=x^2+2x+2x+4\\ &=x^2+4x+4. \end{aligned} \]

The calculation was carried out by applying the distributive law. The formulae known as special products allow certain recurrent expansions to be shortened, but they always derive from this same law.

If \(P\) is a non-zero polynomial and \(m\) is a positive integer, then

\[ \deg(P^m)=m\deg(P). \]

Indeed, \(P^m\) is the product of \(m\) factors all equal to \(P\), and the degree of a product is the sum of the degrees of the individual factors.

For every non-zero polynomial one sets, in addition,

\[ P(x)^0=1. \]

The case in which the polynomial is the zero polynomial and the exponent is \(0\) is left undefined.

The operations just studied show that addition, subtraction and multiplication always produce polynomials; the same holds for powers with positive integer exponent. For every non-zero polynomial the power with exponent \(0\) is defined as well, whereas division requires further conditions and will be treated separately.

Numerical Value of a Polynomial

The numerical value of a polynomial is obtained by substituting given numerical values for its variables and carrying out the indicated operations.

Consider, for example, the polynomial

\[ P(x)=2x^3-5x+1. \]

Assigning the value \(2\) to the variable \(x\), we obtain

\[ P(2)=2\cdot 2^3-5\cdot 2+1. \]

Carrying out the calculations,

\[ P(2)=2\cdot 8-10+1=16-10+1=7. \]

One says accordingly that the numerical value of \(P(x)\) at \(x=2\) is \(7\).

The notation \(P(2)\) denotes the number obtained by substituting \(2\) for every occurrence of the variable \(x\). More generally, if

\[ P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0, \]

then, for a given real number \(c\), we have

\[ P(c)=a_nc^n+a_{n-1}c^{n-1}+\cdots+a_1c+a_0. \]

When the value assigned to the variable is negative, it is important to enclose it in brackets before raising it to a power. Consider, for example,

\[ Q(x)=x^3-2x^2+4. \]

For \(x=-2\) we obtain

\[ Q(-2)=(-2)^3-2(-2)^2+4. \]

Hence

\[ Q(-2)=-8-2\cdot 4+4=-8-8+4=-12. \]

The brackets make it possible to distinguish correctly between the power of a negative number and the opposite of a power. Indeed,

\[ (-2)^2=4, \qquad -2^2=-4. \]

The numerical value may also be computed for polynomials in several variables. Consider

\[ P(x,y)=3x^2y-2xy^2+5. \]

For \(x=2\) and \(y=-1\) we have

\[ P(2,-1)=3\cdot 2^2\cdot(-1)-2\cdot 2\cdot(-1)^2+5. \]

Working out the calculations,

\[ P(2,-1)=3\cdot 4\cdot(-1)-4+5=-12-4+5=-11. \]

In general, if \(P(x_1,x_2,\dots,x_n)\) is a polynomial in \(n\) variables, its numerical value at the real numbers \(c_1,c_2,\dots,c_n\) is denoted by

\[ P(c_1,c_2,\dots,c_n) \]

and is obtained by substituting \(c_i\) for the variable \(x_i\) for each \(i=1,2,\dots,n\).

Since a polynomial involves only sums, differences, products and powers with non-negative integer exponent, its numerical value is defined for every real choice of the variables.

With every polynomial in one variable one may therefore associate a function assigning to each real number \(x\) the corresponding value \(P(x)\):

\[ x\longmapsto P(x). \]

A polynomial and the function associated with it are closely related, but they must not be confused: the former is an algebraic object represented by a formal expression, whereas the latter is the function that assigns to each value of the variable the corresponding numerical value of the polynomial.