Adding and Subtracting Polynomials

A polynomial is an expression made up of terms that are added or subtracted, where each term is a number times a variable raised to a whole-number exponent. Expressions like $3x^2 + 5x - 7$, $4y^3 - y + 2$, and even a plain number like $12$ are all polynomials. They are the workhorses of algebra — you will add, subtract, multiply, and factor them constantly.

This page covers the foundational skills: naming polynomials, identifying like terms, and performing addition and subtraction.

Polynomial Vocabulary

Parts of a Term

Each term in a polynomial has two parts:

• Coefficient: the numerical factor (the number in front). In $7x^3$, the coefficient is 7.
• Variable part: the variable with its exponent. In $7x^3$, the variable part is $x^3$.

A term with no variable (like $-5$) is called a constant term.

Naming by Number of Terms

Degree of a Polynomial

The degree of a term is the exponent on its variable (for multi-variable terms, it is the sum of all exponents). The degree of the polynomial is the highest degree among all its terms.

• $5x^3 - 2x + 9$ has degree 3 (from the $5x^3$ term)
• $4x^2 + 7x^2$ simplifies to $11x^2$, which has degree 2
• The constant $12$ has degree 0

Standard Form

A polynomial is in standard form when its terms are written from highest degree to lowest degree:

$\text{Standard form: } 3x^4 - x^3 + 7x^2 - 2x + 5$

Always write your final answers in standard form.

Like Terms

Like terms have the same variable raised to the same exponent. Only the coefficients can differ.

You can only combine like terms. This is the single most important rule for adding and subtracting polynomials.

How to Combine Like Terms

Add (or subtract) the coefficients while keeping the variable part unchanged:

$3x^2 + 5x^2 = (3 + 5)x^2 = 8x^2$

$7y - 2y = (7 - 2)y = 5y$

Example 1: Simplify $4x^3 + 2x - 7x^3 + 5x + 3$

Group like terms:

$(4x^3 - 7x^3) + (2x + 5x) + 3$

Combine coefficients:

$-3x^3 + 7x + 3$

Answer: $-3x^3 + 7x + 3$

Adding Polynomials

To add polynomials, remove the parentheses and combine like terms. Since you are adding, the signs of all terms stay the same.

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

Step 1 — Remove parentheses (signs stay the same):

$3x^2 + 5x - 4 + 2x^2 - 3x + 7$

Step 2 — Group like terms:

$(3x^2 + 2x^2) + (5x - 3x) + (-4 + 7)$

Step 3 — Combine:

$5x^2 + 2x + 3$

Answer: $5x^2 + 2x + 3$

Example 3: $(x^3 + 4x^2 - x) + (2x^3 - x^2 + 6x - 5)$

Step 1 — Remove parentheses:

$x^3 + 4x^2 - x + 2x^3 - x^2 + 6x - 5$

Step 2 — Group and combine like terms:

$(x^3 + 2x^3) + (4x^2 - x^2) + (-x + 6x) + (-5)$

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

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

Vertical Method

Some students prefer stacking polynomials vertically, aligning like terms in columns:

$\begin{array}{r} 3x^2 + 5x - 4 \\ + \; 2x^2 - 3x + 7 \\ \hline 5x^2 + 2x + 3 \end{array}$

Both methods give the same result. Use whichever feels more natural.

Subtracting Polynomials

Subtracting a polynomial means distributing the negative sign (multiplying each term in the second polynomial by $-1$) and then combining like terms. This is where most mistakes happen.

Example 4: $(6x^2 + 3x - 8) - (2x^2 - 5x + 1)$

Step 1 — Distribute the negative sign to every term in the second polynomial:

$6x^2 + 3x - 8 - 2x^2 + 5x - 1$

Notice how every sign in the second polynomial flipped: $+2x^2$ became $-2x^2$, $-5x$ became $+5x$, and $+1$ became $-1$.

Step 2 — Group and combine like terms:

$(6x^2 - 2x^2) + (3x + 5x) + (-8 - 1)$

$4x^2 + 8x - 9$

Answer: $4x^2 + 8x - 9$

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

Step 1 — Distribute the negative sign:

$5x^3 - 2x + 7 - 3x^3 - x^2 + 4x - 3$

Step 2 — Group and combine like terms:

$(5x^3 - 3x^3) + (-x^2) + (-2x + 4x) + (7 - 3)$

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

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

Example 6: Subtract Using the Vertical Method

Find $(7x^2 + 4x - 3) - (5x^2 - 2x + 6)$.

First, change the signs of the second polynomial, then add:

$\begin{array}{r} 7x^2 + 4x - 3 \\ + \;(-5x^2 + 2x - 6) \\ \hline 2x^2 + 6x - 9 \end{array}$

Answer: $2x^2 + 6x - 9$

Polynomial Addition and Subtraction with Missing Terms

When one polynomial is missing a term of a certain degree, treat that term’s coefficient as zero.

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

The first polynomial has no $x^2$ or $x$ terms. Think of it as $x^3 + 0x^2 + 0x + 5$.

$(x^3 + 2x^3) + (0x^2 - 4x^2) + (0x + x) + 5$

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

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

Real-World Application: Carpentry — Perimeter of a Custom Frame

A carpenter is building a rectangular picture frame with a decorative border. The frame has:

• Length: $(3x + 5)$ inches (where $x$ depends on the customer’s chosen artwork size)
• Width: $(2x - 1)$ inches

The perimeter (total length of molding needed) is:

$P = 2 \times \text{length} + 2 \times \text{width}$

$P = 2(3x + 5) + 2(2x - 1)$

Step 1 — Distribute:

$P = 6x + 10 + 4x - 2$

Step 2 — Combine like terms:

$P = (6x + 4x) + (10 - 2) = 10x + 8$

If the customer’s artwork requires $x = 4$ inches:

$P = 10(4) + 8 = 40 + 8 = 48 \text{ inches} = 4 \text{ feet}$

Answer: The carpenter needs 48 inches (4 feet) of molding. By keeping the expression in polynomial form first, the carpenter has a reusable formula — change $x$ for different artwork sizes without reworking the entire calculation.

Common Mistakes to Avoid

1. Forgetting to distribute the negative sign to every term. When subtracting $(2x^2 - 5x + 1)$, all three signs flip: $-2x^2 + 5x - 1$. The most common error is flipping the first sign but leaving the others unchanged.
2. Combining unlike terms. $3x^2 + 5x$ cannot be simplified further — they have different exponents, so they are not like terms.
3. Losing the sign of a coefficient. Be careful with terms like $-x^2$, which has a coefficient of $-1$, not $0$ or $1$.
4. Not writing the answer in standard form. After combining like terms, arrange from highest degree to lowest.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A polynomial is a sum of terms with whole-number exponents; terms are classified by their degree and polynomials by their number of terms (monomial, binomial, trinomial)
• The degree of a polynomial is the highest exponent — always write the final answer in standard form (highest to lowest degree)
• Like terms have the same variable and exponent — combine them by adding or subtracting their coefficients
• Adding polynomials: Drop the parentheses and combine like terms
• Subtracting polynomials: Distribute the negative sign to every term in the polynomial being subtracted, then combine like terms
• The most common mistake is forgetting to flip all signs when subtracting

Return to Algebra for more topics in this section.