Adding and Subtracting Polynomials

A polynomial is an expression made of variables and coefficients combined with addition, subtraction, and multiplication (no division by a variable). You've already been working with polynomials — expressions like $3x + 5$ and $x^2 - 4x + 7$ qualify.

Learning to add and subtract polynomials is mostly about combining like terms, which you already know how to do. The key is keeping track of multiple variables and different powers.

Polynomial Vocabulary

A few terms worth knowing before getting into the operations.

A term is a single part of a polynomial. In $3x^2 - 5x + 7$, there are three terms: $3x^2$, $-5x$, and $7$.

The coefficient is the number in front of the variable. In $3x^2$, the coefficient is 3.

The degree of a term is the sum of the exponents on its variables. The term $5x^2$ has degree 2. The term $3xy^2$ has degree $1 + 2 = 3$.

The degree of a polynomial is the highest degree of any of its terms. The polynomial $4x^3 - 2x + 1$ has degree 3.

Types of polynomials by number of terms:

Monomial: one term ($5x^2$)
Binomial: two terms ($x + 3$)
Trinomial: three terms ($x^2 + 2x + 1$)
Polynomial: more than three terms, or the general term for all of these

Standard Form

Polynomials are typically written in standard form, with terms arranged from highest degree to lowest degree.

Example: $3 + 5x - 2x^2$ should be written as $-2x^2 + 5x + 3$.

Example: $7x + x^3 - 4x^2 + 1$ in standard form is $x^3 - 4x^2 + 7x + 1$.

Adding Polynomials

To add polynomials, combine like terms. Like terms have the same variable part with the same exponents.

Example: Add $(3x^2 + 5x - 2) + (x^2 - 3x + 7)$.

Remove parentheses (the plus sign doesn't change anything): $$3x^2 + 5x - 2 + x^2 - 3x + 7$$

Group like terms: $$(3x^2 + x^2) + (5x - 3x) + (-2 + 7)$$

Combine: $$4x^2 + 2x + 5$$

Example: $(2y^3 + 4y - 1) + (3y^2 - y + 5)$

Combine like terms (note that $y^3$ and $y^2$ are NOT like terms): $$2y^3 + 3y^2 + 4y - y - 1 + 5$$ $$= 2y^3 + 3y^2 + 3y + 4$$

Subtracting Polynomials

Subtracting is trickier because you need to distribute the negative sign to every term in the second polynomial.

Example: Subtract $(5x^2 + 3x - 4) - (2x^2 - x + 1)$.

Distribute the negative: $$5x^2 + 3x - 4 - 2x^2 + x - 1$$

Notice that $-x$ became $+x$ and $+1$ became $-1$. The minus sign in front of the parentheses flips the sign of every term inside.

Combine like terms: $$3x^2 + 4x - 5$$

Quick check: what does the subtraction $(6x^2 + x - 2) - (6x^2 - x + 2)$ simplify to? Show answerDistribute the negative: $6x^2 + x - 2 - 6x^2 + x - 2$. The $x^2$ terms cancel, leaving $2x - 4$.

Example: $(4a^2 - 2a + 7) - (a^2 + 5a - 3)$

Distribute the negative: $$4a^2 - 2a + 7 - a^2 - 5a + 3$$

Combine: $$3a^2 - 7a + 10$$

Working with Multiple Variables

Polynomials can have more than one variable. The same principles apply: combine like terms, leave unlike terms alone.

Example: Add $(3xy + 2x - y) + (xy - 5x + 4y)$.

$$3xy + 2x - y + xy - 5x + 4y$$

Group like terms: $$(3xy + xy) + (2x - 5x) + (-y + 4y)$$

Combine: $$4xy - 3x + 3y$$

Example: Subtract $(5x^2 y - 3xy + 2) - (2x^2 y + xy - 1)$.

Distribute the negative: $$5x^2 y - 3xy + 2 - 2x^2 y - xy + 1$$

Combine: $$3x^2 y - 4xy + 3$$

Adding and Subtracting Vertically

You can also align polynomials vertically, similar to adding numbers. This helps keep like terms organized.

Example: Add $4x^2 + 3x - 5$ and $2x^2 - x + 7$.

Align by degree: $$\begin{array}{r} 4x^2 + 3x - 5 \

\ 2x^2 - x + 7 \ \hline 6x^2 + 2x + 2 \end{array}$$

Example: Subtract $6y^3 + 2y - 3$ from $9y^3 - 5y + 1$.

$$\begin{array}{r} 9y^3 + 0y^2 - 5y + 1 \

(6y^3 + 0y^2 + 2y - 3) \ \hline 3y^3 + 0y^2 - 7y + 4 \end{array}$$

Note the $0y^2$ placeholder added to keep the columns aligned by degree.

A More Complex Example

Example: Simplify $(2x^3 - x^2 + 4x) + (x^2 - 5x + 3) - (x^3 + 2x - 1)$.

Remove the first set of parentheses: $$2x^3 - x^2 + 4x + x^2 - 5x + 3 - (x^3 + 2x - 1)$$

Distribute the negative on the last set: $$2x^3 - x^2 + 4x + x^2 - 5x + 3 - x^3 - 2x + 1$$

Combine like terms: $$(2x^3 - x^3) + (-x^2 + x^2) + (4x - 5x - 2x) + (3 + 1)$$ $$= x^3 - 3x + 4$$

Try These

Add: $(5x^2 + 3x - 1) + (2x^2 - x + 4)$ Show answerCombine like terms: $(5x^2 + 2x^2) + (3x - x) + (-1 + 4) = 7x^2 + 2x + 3$
Subtract: $(7y^2 - 4y + 2) - (3y^2 + y - 5)$ Show answerDistribute the negative: $7y^2 - 4y + 2 - 3y^2 - y + 5 = 4y^2 - 5y + 7$
Add: $(4ab + 2a - b) + (3ab - a + 3b)$ Show answerCombine like terms: $(4ab + 3ab) + (2a - a) + (-b + 3b) = 7ab + a + 2b$
Simplify: $(x^3 + 2x - 5) + (3x^2 - x + 1) - (x^3 - x^2 + 4)$ Show answerDistribute the negative on the last group: $x^3 + 2x - 5 + 3x^2 - x + 1 - x^3 + x^2 - 4$. The $x^3$ terms cancel. Combine: $4x^2 + x - 8$
Subtract: $(9m^2 - 3m + 7) - (5m^2 - 2m - 1)$ Show answerDistribute the negative: $9m^2 - 3m + 7 - 5m^2 + 2m + 1 = 4m^2 - m + 8$

What's Next?

A few habits keep these problems clean. The most common mistake is forgetting that a minus sign in front of parentheses flips every term inside — subtracting $(3y^2 + y - 5)$ means subtracting $3y^2$, subtracting $y$, and adding 5 (because subtracting $-5$ is adding). When combining terms, only true like terms merge: $x^2$ and $x$ stay separate (different exponents), and $xy$ and $x$ stay separate (different variable parts). When you do combine, you add the coefficients only — the exponent stays the same: $3x^2 + 2x^2 = 5x^2$, never $5x^4$. And finish in standard form, with terms arranged from highest degree to lowest.

Once you're comfortable adding and subtracting, the next step is multiplying polynomials, which builds on the distributive property to combine entire polynomials term by term.