Add, subtract, and multiply SAT polynomials, combine like terms carefully, and use the factor and remainder theorems to test divisibility.
Add coefficients of terms with the same variable and power
An $x^2$ term can never combine with an $x$ term or a constant.
The minus sign flips every term.
Do not forget the middle term $2ab$.
Remainder of is ; is a factor iff
A polynomial is a sum of terms, each a number times a power of a variable, such as . Polynomial arithmetic — adding, subtracting, and multiplying these expressions — behaves just like arithmetic with numbers, governed by a single rule: you can only combine like terms, terms with the exact same variable raised to the exact same power. The SAT tests this constantly, usually hiding a small sign error where a careless student loses an easy point.
To add or subtract polynomials, combine like terms and leave unlike terms alone. Line up matching powers:
because , , and . Subtraction carries one extra danger: the minus sign distributes to every term in the second polynomial.
Watch how became and became . Forgetting to flip every sign inside the parentheses is the number-one SAT error on these problems.
To multiply, distribute every term of the first polynomial across every term of the second, then combine like terms. For two binomials this is the familiar FOIL pattern, but the real rule is simply "each times each":
For larger products the same principle scales up: every term in the left factor multiplies every term in the right, and then you gather like powers.
Two products appear so often they are worth recognizing on sight:
The first shows why has a middle term — a frequent trap is writing and forgetting the . The second, difference of squares, is the reverse of a factoring pattern and lets you multiply conjugates instantly.
Expand and simplify .
Combining, and . Organizing the work by which term you distributed keeps the like terms easy to line up.
The SAT connects arithmetic to factoring through two clean facts about dividing a polynomial by :
Read this both directions: a zero output means a factor, and a nonzero output is the remainder. This shortcut turns many "is this a factor?" or "find the remainder" questions into a single substitution.
The factor theorem also solves for unknown constants. If is a factor of , then :
Substituting the root that the factor announces, then solving the resulting linear equation, is a signature Advanced Math move.
Polynomial arithmetic is not conceptually hard, but it punishes sloppiness. Distribute every subtraction sign, keep only like terms together, track each term in a long multiplication, and remember that a factor produces a zero — not a large — output. Neatness here is worth real points.
Simplify .
Simplify .
Expand .
Expand and simplify .
What is the remainder when is divided by ?
If is a factor of , what is the value of ?
Distributing a subtraction to only the first term, e.g. .
The minus sign hits every term: . Flip all three signs.
Combining unlike terms, such as adding an term to an term.
Only terms with the same variable and exponent are alike. and combine; and do not.
Writing , forgetting the middle term.
Use : . A squared binomial always has three terms.
Thinking a large value of means is a factor.
is a factor only when . Any nonzero value of is the remainder, not evidence of a factor.
What is written in simplest form?
The polynomial has as a factor. What is ?
What is in simplest form?
Which expression is equivalent to ?
Expand and simplify .
What is the remainder when is divided by ?
Factor SAT polynomials using GCF, difference of squares, trinomials, and grouping — then use the factored form to solve equations, find zeros, and simplify expressions.
Solve SAT quadratic equations with factoring, the square-root method, completing the square, and the quadratic formula — and use the discriminant to count real solutions.