Polynomial Operations on the SAT
Expand, Combine, and Compare
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Polynomial Operations
Polynomial operations show up when the SAT asks you to expand, simplify, or compare expressions. These questions are less about clever tricks and more about careful organization.
This lesson shows how to distribute cleanly, combine like terms, and check your work with Desmos if needed. You will see why missing a single term can change the entire answer.
A Simple Definition Unlocks Polynomial Operations
To multiply polynomials, distribute each term in the first expression across the second. For binomials, many students use FOIL, but the same logic works for any size polynomial.
After you distribute, combine like terms by adding their coefficients. A quick scan for missing degrees, like forgetting the $x^2$ term, helps you avoid the most common errors.
Work Through Polynomial Operations Step by Step
Expand and simplify $(x + 2)(x - 5)$.
Multiply and simplify a polynomial product so every term appears once.
Start with the product before distributing.
Distribute across the parentheses to expand the expression.
Combine like terms to simplify the expression.
Use Desmos to Check Polynomial Operations
Expand and simplify $(x + 2)(x - 5)$.
Desmos can quickly spot a mismatch by graphing two expressions, but it does not prove equivalence. Use it to check a few values or see if graphs differ, then rely on algebra for the exact expansion.
y = (x + 2)(x - 5)
y = x^2 - 3x - 10
Algebra is still the main method for exact expansion and simplification. Desmos is a fast check when expressions are long or the answer choices look similar.
Expert move: Use Desmos to disprove an answer quickly - if the graphs or table values differ at any $x$, the expressions are not equivalent.
Verification tip: If graphs overlap, still do the algebra or test a few x-values to confirm the exact expansion.
- Desmos features used: graphing, overlay comparison.
- Common mistake: missing a term when distributing.
Practice Polynomial Operations with SAT-Style Questions
Expand, simplify, or find remainders.
Expand .
Simplify .
Find the remainder when is divided by .
Which expression is equivalent to ?
Key Takeaways to Remember for Polynomial Operations
- Distribute carefully and combine like terms.
- The Remainder Theorem lets you plug in $x = a$ for division by $(x - a)$.
- Desmos is a fast way to compare polynomial equivalence.
About the Instructor
