Adding and Subtracting Polynomials

A polynomial is an expression made up of variables and coefficients, using only addition, subtraction, and multiplication (no division by variables). You've been working with polynomials already—things like $3x + 5$ or $x^2 - 4x + 7$ are polynomials.

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

First, let's establish some terminology.

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: $$3x^2 + 4x - 5$$

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.

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}$$

Notice I included $0y^2$ as a placeholder to keep columns aligned.

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: $(5x^2 + 3x - 1) + (2x^2 - x + 4)$
Subtract: $(7y^2 - 4y + 2) - (3y^2 + y - 5)$
Add: $(4ab + 2a - b) + (3ab - a + 3b)$
Simplify: $(x^3 + 2x - 5) + (3x^2 - x + 1) - (x^3 - x^2 + 4)$
Subtract: $(9m^2 - 3m + 7) - (5m^2 - 2m - 1)$

Solutions:

$7x^2 + 2x + 3$
$7y^2 - 4y + 2 - 3y^2 - y + 5 = 4y^2 - 5y + 7$
$7ab + a + 2b$
$x^3 + 2x - 5 + 3x^2 - x + 1 - x^3 + x^2 - 4 = 4x^2 + x - 8$
$9m^2 - 3m + 7 - 5m^2 + 2m + 1 = 4m^2 - m + 8$

Things to Watch For

Forgetting to distribute the negative sign when subtracting is the number one mistake. When you subtract $(3y^2 + y - 5)$, that minus sign flips every single term: you're subtracting $3y^2$, subtracting $y$, and subtracting $-5$ (which means adding 5).

Trying to combine unlike terms shows you don't understand what "like" means. You cannot combine $x^2$ and $x$ because they have different exponents. You cannot combine $xy$ and $x$ because they have different variable parts. Keep them separate.

Losing track of signs, especially with multiple negatives, leads to arithmetic errors. When you see $-5x - (-3x)$, remember that subtracting a negative is the same as adding: $-5x + 3x = -2x$.

Not arranging your final answer in standard form (highest degree to lowest degree) looks sloppy and makes the answer harder to read. $3 + 5x - 2x^2$ should be written as $-2x^2 + 5x + 3$.

Combining coefficients incorrectly is usually just careless arithmetic. In $3x^2 + 2x^2$, you add the coefficients ($3 + 2 = 5$) to get $5x^2$. You don't add the exponents—that's a completely different operation.