Add and Subtract Polynomials

Identify polynomials, monomials, binomials, and trinomials

You have learned that a term is a constant or the product of a constant and one or more variables. The constant is called a coefficient. When a term is of the form $ax^m$, where $a$ is a constant and $m$ is a whole number, it is called a monomial. Some examples of monomials are $8$, $-2x^2$, $4y^3$, and $11z^7$.

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms:

• A monomial is a polynomial with exactly one term.
• A binomial is a polynomial with exactly two terms.
• A trinomial is a polynomial with exactly three terms.

There are no special names for polynomials with more than three terms. Here are some examples of each kind:

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials, and so they have special names.

Example. Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

(a) $4y^2 - 8y - 6$ has three terms, so it is a trinomial.

(b) $-5a^4 b^2$ has one term, so it is a monomial.

(c) $2x^5 - 5x^3 - 9x^2 + 3x + 4$ has five terms, so it is a polynomial.

(d) $13 - 5m^3$ has two terms, so it is a binomial.

(e) $q$ has one term, so it is a monomial.

Determine the degree of polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable. A monomial that has no variable, just a constant, is a special case: the degree of a constant is $0$ — it has no variable.

For example, the monomial $-9x^4 y^6$ has degree $4 + 6 = 10$. In the binomial $3n^3 - 9n$, the terms have degrees $3$ and $1$, so the polynomial has degree $3$. In the trinomial $6m^4 - m^3 n^2 + 8mn^5$, the terms have degrees $4$, $5$, and $6$, so the polynomial has degree $6$.

A polynomial is in standard form when its terms are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

Example. Find the degree of each polynomial.

(a) $10y$ — the exponent of $y$ is one, since $y = y^1$, so the degree is $1$.

(b) $4x^3 - 7x + 5$ — the highest degree of all the terms is $3$, so the degree is $3$.

(c) $-15$ — the degree of a constant is $0$.

(d) $-8b^2 + 9b - 2$ — the highest degree of all the terms is $2$, so the degree is $2$.

(e) $8xy^2 + 2y$ — the term $8xy^2$ has degree $1 + 2 = 3$, so the degree is $3$.

Add and subtract monomials

You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

Example. Add: $25y^2 + 15y^2$.

The two terms are like terms, so we combine them by adding the coefficients:

Example. Subtract: $16p - (-7p)$.

Subtracting a negative is the same as adding, so we combine like terms:

Example. Simplify: $c^2 + 7d^2 - 6c^2$.

Only $c^2$ and $-6c^2$ are like terms. Combining them:

Example. Simplify: $u^2 v + 5u^2 - 3v^2$.

There are no like terms to combine, so the expression stays as it is:

Add and subtract polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms — those with the same variables and the same exponent. The Commutative Property lets us rearrange the terms to put like terms together.

Example. Find the sum: $\left(5y^2 - 3y + 15\right) + \left(3y^2 - 4y - 11\right)$.

Rearrange to group the like terms, then combine them:

Example. Find the difference: $\left(9w^2 - 7w + 5\right) - \left(2w^2 - 4\right)$.

Distribute the subtraction across the second polynomial, rearrange, and combine like terms:

Example. Subtract $\left(c^2 - 4c + 7\right)$ from $\left(7c^2 - 5c + 3\right)$.

“Subtract $A$ from $B$” means $B - A$, so we set up the difference and distribute:

Polynomials in more than one variable work the same way — we just have to be careful to combine only genuine like terms.

Example. Find the sum: $\left(u^2 - 6uv + 5v^2\right) + \left(3u^2 + 2uv\right)$.

Example. Find the difference: $\left(p^2 + q^2\right) - \left(p^2 + 10pq - 2q^2\right)$.

Evaluate a polynomial for a given value

We have already learned how to evaluate expressions. Since polynomials are expressions, we follow the same procedure: substitute the given value for the variable and then simplify using the order of operations.

Example. Evaluate $5x^2 - 8x + 4$ when (a) $x = 4$, (b) $x = -2$, and (c) $x = 0$.

(a) Substitute $4$ for $x$:

(b) Substitute $-2$ for $x$:

(c) Substitute $0$ for $x$:

Example. The polynomial $-16t^2 + 250$ gives the height (in feet) of a ball $t$ seconds after it is dropped from a $250$-foot tall building. Find the height after $t = 2$ seconds.

Substitute $t = 2$ and simplify:

After $2$ seconds the height of the ball is $186$ feet.

Example. The polynomial $6x^2 + 15xy$ gives the cost (in dollars) of producing a rectangular container whose top and bottom are squares with side $x$ feet and sides of height $y$ feet. Find the cost of producing a box with $x = 4$ feet and $y = 6$ feet.

Substitute $x = 4$ and $y = 6$, then simplify:

The cost of producing the box is $\text{\textdollar}456$.

Key terms

monomial — a term of the form $ax^m$, where $a$ is a constant and $m$ is a positive whole number. polynomial — a monomial, or two or more monomials combined by addition or subtraction. binomial — a polynomial with exactly two terms. trinomial — a polynomial with exactly three terms. coefficient — the constant factor in a term. degree of a term — the sum of the exponents of its variables. degree of a polynomial — the highest degree of all its terms. standard form — a polynomial written with its terms in descending order of degree.

This section is adapted from Elementary Algebra 2e, Section 6.1: Add and Subtract Polynomials by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the polynomial-classification and degree examples as prose with (a)/(b)/(c) enumerations and a summary table, and the worked add/subtract examples as display equality chains; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback — using multiple-choice for the “name the type of polynomial” Try Its, since a word answer can’t be graded by the math checker.