Lesson 1: Add and Subtract Polynomials

New Concept

This lesson introduces polynomials: algebraic expressions like monomials and trinomials. You'll learn to classify them by their number of terms and degree, and then add and subtract them by combining like terms—a foundational algebraic skill.

What’s next

You've got the foundation! Next, you’ll tackle practice cards and worked examples to build your skills in adding and subtracting polynomials.

Property

A monomial, or a sum and/or difference of monomials, is called a polynomial.
polynomial—A monomial, or two or more monomials, combined by addition or subtraction
monomial—A polynomial with exactly one term
binomial—A polynomial with exactly two terms
trinomial—A polynomial with exactly three terms

Examples

• The expression $10x^3 - 4x + 1$ has three terms, so it is a trinomial.
• The expression $9y^5$ has only one term, making it a monomial.
• The expression $a^2 - 16$ has two terms, which means it is a binomial.

Explanation

Think of the name as a clue! The prefixes tell you the number of terms: 'mono-' for one, 'bi-' for two, and 'tri-' for three. A polynomial is the family name for all of them.

Property

The degree of a term is the exponent of its variable.
The degree of a constant is 0.
The degree of a polynomial is the highest degree of all its terms.
When a polynomial's terms are listed in descending order of degrees, it is in standard form.

Examples

• In the polynomial $8x^4 - 2x^2 + 5$, the degrees of the terms are 4, 2, and 0. The highest degree is 4, so the degree of the polynomial is 4.
• The degree of the monomial $-15p^3$ is 3, because the exponent of the variable $p$ is 3.
• The degree of the constant term $25$ is 0, as it has no variable.

Explanation

To find a polynomial's degree, just look for the term with the highest exponent—that single number is the degree!

Property

Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. When combining like terms, only the coefficients are combined, never the exponents.

Examples

• To add $8y^3 + 5y^3$, we combine the coefficients since the variable and exponent match: $(8+5)y^3 = 13y^3$.
• To subtract $15m - (-4m)$, we combine the coefficients: $(15 - (-4))m = 19m$.
• The terms $9x^2$ and $2y^2$ are not like terms because their variables are different, so their sum is simply written as $9x^2 + 2y^2$.

Explanation

You can only combine terms that are exactly alike—same variable and same exponent.

Property

Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms—those with the same variables with the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Examples

• To find the sum $(3x^2 - 4x + 8) + (5x^2 + 2x - 3)$, combine like terms: $(3x^2+5x^2) + (-4x+2x) + (8-3) = 8x^2 - 2x + 5$.
• To find the difference $(8y^2 + 6y - 5) - (3y^2 - y + 7)$, distribute the negative: $8y^2 + 6y - 5 - 3y^2 + y - 7$, which simplifies to $5y^2 + 7y - 12$.
• Subtract $(p^2 - 4p + 1)$ from $(10p^2 + 2p - 5)$: $(10p^2 + 2p - 5) - (p^2 - 4p + 1) = 10p^2 + 2p - 5 - p^2 + 4p - 1 = 9p^2 + 6p - 6$.

Explanation

To add polynomials, just group and combine the like terms. For subtraction, first distribute the negative sign to every term in the second polynomial, then combine the like terms as you would with addition.