Introduction
Polynomial operations sit squarely in the Advanced Math domain, which makes up roughly 35% of SAT Math questions. You will be asked to simplify, expand, or match an expression, often to find a specific coefficient or constant term. The skill appears in both straightforward multiple-choice questions and in student-produced response (SPR) questions, usually in 3–5 questions per test.
Core Concept
A polynomial is a sum of terms of the form where is a real number and is a non-negative integer. Polynomial operations have three moves:
1. Adding and Subtracting — Combine Like Terms
Terms are like if they share the same variable and exponent. Add or subtract their coefficients; leave the variable part alone.
2. Distributing a Negative — The #1 Error Source
Subtracting a polynomial means distributing the negative to every term inside the parentheses.
The signs of , , and all flipped. Miss one flip and you get a wrong answer — SAT distractors are built exactly on this error.
3. Multiplying Polynomials — Distribute Every Term
Use the distributive property: multiply each term in the first polynomial by each term in the second.
Binomial × binomial (FOIL is just a memory device for this case):
Larger products — the same rule, more pairs:
Missing Coefficients
A common SAT format gives you an expanded form with one coefficient labeled as an unknown variable (say ), then asks for . Expand the product fully, collect like terms, then match coefficients term-by-term.
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