Lesson 11: Understanding Polynomials

New Concept

A polynomial is a monomial or sum of monomials. The terms of a polynomial are its monomials.

What’s next

Next, you’ll learn to classify, add, and subtract these expressions to build your foundational algebraic skills.

The degree of a polynomial is the degree of its monomial with the greatest degree. The degree of a monomial is the sum of the exponents of its variable factors.

The degree of $7x^2y^4$ is $2+4=6$.
For $-2a^2b^3 + 4a^4b - 9a^3$, the term degrees are 5, 5, and 3. The polynomial degree is 5.
The polynomial $x^3 + x^7 - 2x^2$ has degree 7.

Think of it like a superhero team's power level! To find a polynomial's degree, first find the power level of each term by adding up the exponents on its variables. The biggest number you find is the polynomial's official degree, its highest rank. This tells you which term is the most powerful in the group.

A one-variable polynomial is in standard form when all like terms have been combined and its terms are in descending order by degree.

$x - 8 + 5x^3 - x^2$ in standard form is $5x^3 - x^2 + x - 8$.
In standard form, $6y + 2y^4 - 3y^2$ becomes $2y^4 - 3y^2 + 6y$. The leading coefficient is 2.
Combining terms in $z + 7z^2 - 3 - 2z^2$ gives $5z^2 + z - 3$. The constant term is $-3$.

Putting a polynomial in standard form is like organizing your bookshelf by the height of the books. You place the term with the biggest exponent first, then the next biggest, and so on, all the way down to the plain number. This tidies it up and makes spotting the leader and caboose super easy.

Polynomials are classified by their degree and by their number of terms. Common degree names are quadratic (degree 2) and cubic (degree 3). Common term names are monomial (1 term), binomial (2 terms), and trinomial (3 terms).

$x^2 - 16$ is a quadratic binomial because its degree is 2 and it has 2 terms.
The polynomial $-5x^4 + x^3 - 8x$ is a quartic trinomial (degree 4, 3 terms).
The term $10y^3$ is a cubic monomial since its degree is 3 and it is a single term.

Just like we classify animals, we classify polynomials to understand them better. First, we look at their highest degree to name their 'family' (like cubic or quadratic). Then, we count how many terms they have to give them a second name (like binomial or trinomial). It's a fun, two-part naming system for math expressions!

To add polynomials, combine like terms. To subtract a polynomial, add the opposite of each of its terms.

Add: $(3x^2 + 2x - 1) + (x^2 - 5x + 6) = 3x^2 + x^2 + 2x - 5x - 1 + 6 = 4x^2 - 3x + 5$.
Subtract: $(4y^3 - y) - (2y^3 + 3y - 2) = 4y^3 - y - 2y^3 - 3y + 2 = 2y^3 - 4y + 2$.

Adding and subtracting polynomials is all about matching! You can only combine 'like terms'—those with the exact same variables and exponents. When you subtract, you must first flip the sign of every single term in the second polynomial. Then it becomes a simple game of sorting, matching, and combining the similar pieces together.