Polynomial Arithmetic (Add, Subtract, Multiply) on the Digital SAT

Quick Answer: Polynomial arithmetic involves adding, subtracting, and multiplying algebraic expressions by combining like terms and using the distributive property. On the Digital SAT, pay special attention to distributing negative signs when subtracting, and remember you can use the Desmos calculator to verify equivalent expressions by graphing them.

graph TD
    A[See Polynomial Expression] --> B{Operation?}
    B -->|Addition| C[Identify and combine like terms]
    B -->|Subtraction| D[Distribute the negative sign, then combine like terms]
    B -->|Multiplication| E[Distribute every term to every other term, then combine]

What Is Polynomial Arithmetic?

Polynomial arithmetic is the foundation of many Advanced Math questions on the Digital SAT. It involves performing basic operations—addition, subtraction, and multiplication—on polynomials. Addition and subtraction require you to identify and combine "like terms" (terms with the exact same variable and exponent). Multiplication requires you to use the distributive property to multiply every term in the first polynomial by every term in the second polynomial.

According to the official specifications from the College Board, mastering these operations is essential for the 2026 Digital SAT format. You will frequently need to simplify expressions before you can solve an equation, transition into /sat/math/standard-form-quadratic, or prepare an expression for /sat/math/factoring-quadratics.

Fortunately, because the Digital SAT includes a built-in Desmos Calculator, you have powerful visual tools at your disposal to verify your algebraic work.

Step-by-Step Method

1. Step 1 — Identify the operation required (addition, subtraction, or multiplication).
2. Step 2 — If subtracting, distribute the negative sign to every term inside the parentheses of the second polynomial.
3. Step 3 — If multiplying, distribute each term from the first polynomial to every term in the second polynomial (using FOIL for binomials).
4. Step 4 — Group the like terms together (e.g., put all the $x^2$ terms together, all the $x$ terms together, and all the constants together).
5. Step 5 — Combine the coefficients of the like terms to write the final simplified expression.

Desmos Shortcut

If a Digital SAT question asks "Which of the following expressions is equivalent to the expression above?", you can skip the algebra entirely using Desmos. Simply type the original polynomial expression into line 1 of Desmos (e.g., y = (x-2)(x+3)). Then, type the answer choices into lines 2, 3, 4, and 5. The correct answer choice will graph a curve that perfectly overlaps the original curve. If you turn the answer choice graph on and off, the color should toggle right on top of the original graph.

Worked Example

Question: Which of the following expressions is equivalent to $2x(x - 3) - (x^2 - 5x + 2)$?

A) $x^2 - 11x - 2$ B) $x^2 - x - 2$ C) $3x^2 - 11x + 2$ D) $x^2 - x + 2$

Solution:

First, distribute the $2x$ into the first set of parentheses: $2x(x) - 2x(3) = 2x^2 - 6x$

Next, distribute the negative sign into the second set of parentheses (this is where many students make a mistake!): $-1(x^2) - 1(-5x) - 1(2) = -x^2 + 5x - 2$

Now, write out the full expanded expression: $2x^2 - 6x - x^2 + 5x - 2$

Group and combine the like terms: $(2x^2 - x^2) + (-6x + 5x) - 2$

$x^2 - x - 2$

The correct answer is B.

Common Traps

1. The Subtraction Sign Trap — Based on Lumist student data, 15% of algebraic errors involve forgetting to distribute negative signs across parentheses. When you see a minus sign before a polynomial, treat it as a $-1$ and multiply it by every term inside.

2. Mixing Up Addition and Multiplication Rules — Students frequently add exponents when they should only be adding coefficients. For example, $x^2 + x^2 = 2x^2$, but students often mistakenly write $x^4$. Our data shows Advanced Math has a 24% overall error rate, and these foundational arithmetic missteps are a major contributor.

FAQ

How do I multiply two polynomials?

To multiply two polynomials, distribute every term in the first polynomial to every term in the second polynomial. For binomials, the FOIL (First, Outer, Inner, Last) method is a helpful way to remember this process.

Can I use Desmos for polynomial questions?

Yes! If a question asks which expression is equivalent to a given polynomial, you can graph the original expression and the answer choices in Desmos. The correct answer choice will graph exactly on top of the original expression.

What are "like terms" in polynomials?

Like terms are terms that have the exact same variables raised to the exact same powers. For example, $3x^2$ and $5x^2$ are like terms and can be added to make $8x^2$, but $3x^2$ and $5x$ cannot be combined.

How many Polynomial Arithmetic questions are on the SAT?

Advanced Math makes up approximately 35% of SAT Math, and polynomial operations are foundational to many of these problems. On Lumist.ai, we have 30 practice questions specifically on this topic to help you prepare.