How monomials and polynomials work

We need the area of a rectangular plot with sides $(x+2)$ m and $(x+3)$ m. Why does $(x+2)(x+3)$ give $x^2+5x+6$ , rather than just $x^2+6$ ? Because each part of one side forms a rectangle with each part of the other. The two mixed products have areas too.

We will use variables and the distributive law and exponent rules. Here $x$ is the numerical value of a base length in metres; for the visible partition we take $x>0$ .

Four parts instead of a guess

Split the width into $x$ and 2, and the height into $x$ and 3. We get a square of area $x^2$ , two strips of areas $2x$ and $3x$ , and a corner rectangle of area 6. All areas are in square metres.

The drawing uses x = 4: width 6 m, height 7 m, part areas 16, 8, 12 and 6 m². Their sum is 42 m². Letters show the general calculation; this ordinary area partition requires x > 0.

Adding the contributions gives $x^2+2x+3x+6=x^2+5x+6$ . To understand which parts can be combined, let us name the expression’s building blocks.

Monomial, coefficient and degree

A monomial is a number or a product of a numerical coefficient and variables raised to nonnegative integer powers. For example, $-3a^2b$ has coefficient −3 and degree $2+1=3$ . In standard form, numerical factors have already been multiplied and each variable appears as one power: $2a\cdot3ab=6a^2b$ .

A polynomial is a sum of monomials; a minus sign can be viewed as a negative coefficient. Coefficients may be integers, fractions or other real numbers. $x^{-1}=1/x$ is not a monomial of this kind: its exponent −1 is not a nonnegative integer.

A nonzero constant such as 6 has degree 0. We do not assign a degree to the zero polynomial here. The degree of a nonzero polynomial is the greatest degree of its nonzero terms after simplification. For $x^2+5x+6$ it is 2, while $3x^2-3x^2+5=5$ has degree 0. Definitions and operations in OpenStax, §1.4.

Which terms are alike?

$2x$ and $3x$ have the same variable part, so $2x+3x=(2+3)x=5x$ . These are like terms. The powers in $x^2$ and $x$ differ, so we cannot combine their coefficients into one term. Checking $x=2$ quickly exposes that mistake: $x^2+x=6$ , whereas $2x^2=8$ .

To put a polynomial in one variable into standard form, combine like terms, remove zero terms and arrange powers in descending order. For instance, $6+3x+x^2+2x=x^2+5x+6$ .

Addition and subtraction preserve signs

To add, remove the parentheses and collect matching powers:

(x^2+2x-1)+(3x^2-x+4)=4x^2+x+3.

To subtract, change the sign of every term in the second polynomial:

(x^2+2x-1)-(3x^2-x+4)=-2x^2+3x-5.

We are multiplying the entire second parenthesis by −1: $-(3x^2-x+4)=-3x^2+x-4$ . Changing only the first sign would produce a different expression.

Multiplication: every term with every term

The distributive law gives two successive steps:

(x+2)(x+3)=x(x+3)+2(x+3).

Expand both parentheses: $x^2+3x+2x+6=x^2+5x+6$ . Multiplying two binomials initially produces four products; some then become like terms. The same method handles more terms: no contribution can be omitted.

At $x=4$ , the original plot has area $6\cdot7=42$ m². The transformed expression gives $16+20+6=42$ m². A numerical check helps find mistakes; the distributive law justifies the identity for every number.

Branch · Model limitsMore about negative values; the multiplication method is already established.

When the rectangle stops explaining the expression

At $x=-1$ , the algebraic identity still gives $(-1+2)(-1+3)=2$ and $1-5+6=2$ . But we cannot split a side into an ordinary segment of length −1 m. Our drawing of four ordinary areas no longer applies, even if the outer sides remain positive. We then rely on the law of arithmetic rather than a literal geometric picture.

A polynomial keeps track of separate contributions to a result. Like terms can be collected; multiplication must include every pair. That is why the area of our plot includes the middle term $5x$ .