Polynomials and Factoring

Vocabulary practice cards covering polynomial classification, operations with monomials, and various factoring techniques including GCF, grouping, trinomials, and decomposition.

Term

A number or a product of a number with one or more variables, which can be raised to a power (e.g., $5$, $-2x^3$, $3x^2y^2z^3$).

Polynomial

A term or sum of terms in which all variables have whole number exponents and variables only appear in the numerator.

Monomial

A polynomial that contains exactly one term (e.g., $3$, $2x^2y$, $-3a$).

Binomial

A polynomial that contains exactly two terms (e.g., $x+2$, $2xy+3$, $x^2-y$).

Trinomial

A polynomial that contains exactly three terms (e.g., $3x^2+2x-3$, $\sqrt{2}x+y-z$).

Leading Term

The term in a polynomial that has the highest degree.

Degree of a Term

The sum of the exponents of the variables contained within that individual term.

Degree of a Polynomial

The degree of the term in the polynomial that has the highest degree.

Constant

A term in a polynomial that contains no variables and has a degree of $0$.

Leading Coefficient

The numerical coefficient that multiplies the variable(s) in the leading term of a polynomial.

Like Terms

Any terms in a polynomial that have the exact same variables raised to the same power, which can be combined by adding or subtracting their coefficients.

Multiplying Monomials

A two-step procedure: Step 1 is to multiply the constants together, and Step 2 is to multiply the variables together by adding the exponents of variables sharing the same base.

Factoring a Polynomial

The opposite process of multiplying; rewriting a polynomial expression as a product of factors.

Common Factor (GCF)

The largest factor that is the same in each term of a polynomial.

Factoring by Grouping

A factoring method consisting of three steps: Step 1 is to group the polynomial terms in pairs of two; Step 2 is to factor each pair of two; Step 3 is to remove the common factor.

Descending Order

An arrangement where the terms of a polynomial are written in order from the term with the greatest exponent to the term with the least exponent.

Factoring $x^2 + bx + c$ ($a = 1$)

A method where you find two numbers that multiply to equal the $c$ term and add to equal the $b$ term, then factor into two binomials placed as $(x + p)(x + q)$.

Decomposition

The process used to factor trinomials of the form $ax^2 + bx + c$ when $a \neq 1$ (and cannot be factored out) by finding two numbers that multiply to equal $ac$ and add to equal $b$, splitting the middle term, and factoring by grouping.

Difference of Squares

A binomial where a squared term is subtracted from another squared term, factoring into conjugate pairs of the form $(u - v)(u + v)$ (e.g., $25s^2 - 4t^2 = (5s - 2t)(5s + 2t)$).

Sum of Squares

An expression in the form of a sum of two square terms, such as $x^2 + 81$, which is not factorable over real numbers.