Introduction

Every SAT® Math test features polynomials. Therefore, being fluent in operations with polynomials (adding, subtracting, multiplying, and factoring) saves precious minutes. This guide explains the core moves, shows step-by-step examples, and provides quick strategies so that rewriting polynomials feels automatic on test day.

Why Polynomials Matter on the SAT®

Roughly one out of every six SAT® Math questions relies on polynomial structure. Sometimes you must identify an equivalent expression; other times you must simplify a rational expression or solve an equation. However, the underlying skill is the same: recognize patterns and use algebraic properties to rewrite expressions quickly.

Foundations First: Polynomial Vocabulary

A. What counts as a polynomial?

A polynomial is a sum of terms where each term is a constant multiplied by a non-negative integer power of the variable. No negative or fractional exponents appear.

B. Key parts

• Degree – highest exponent
• Term – each separated piece (e.g., 4x^2)
• Coefficient – a number multiplying the variable
• Constant – a term with no variable

C. Leading coefficient & standard form

In standard form, terms are written from highest to lowest degree. The coefficient in front of the first (highest-degree) term is the leading coefficient.

Mini-Check

Identify the parts of 3x^2-5x+7:

• Degree: 2
• Leading coefficient: 3
• Constant: 7
• Middle term coefficient: −5

Basic Operations with Polynomials

A. Adding & Subtracting

Always line up like terms—terms with the same variable and exponent.

Example

Add (4x^2+3x-2)+(2x^2-5x+6).

Step 1: Combine x^2 terms:

4x^2+2x^2=6x^2

Step 2: Combine x terms:

3x-5x=-2x

Step 3: Combine constants:

-2+6=4

Therefore, the sum is 6x^2-2x+4.

B. Multiplying

i. Distributive Property (monomial × polynomial)

Multiply 3x(2x^2-5x+4).

Multiply 3x by every term in the parentheses.

3x(2x^2)+3x(-5x)+3x(4)=6x^3-15x^2+12x

ii. Multiply binomials by multiplying every term in one set of parentheses by every term in the other set

Step	Reason
(x + 2)(x - 3)	Original expression
x(x - 3) + 2(x - 3)	Use distributive property
x^2 - 3x + 2x - 6	Distribute both terms
x^2 - x - 6	Use the distributive property

iii. Special Products

• Square of a binomial: (a+b)^2=a^2+2ab+b^2

• Difference of squares: (a+b)(a-b)=a^2-b^2

Example

Compute (x+7)^2.

Using the formula: x^2+2(7)(x)+7^2=x^2+14x+49

Result: 4x^2-28x+49

Factoring: Turning a Mess into a Product

A. Factoring out the Greatest Common Factor (GCF)

Always look for a GCF first; doing so often simplifies further steps.

Example

Factor 12x^3-18x^2+6x.

The GCF is 6x. Therefore, 12x^3-18x^2+6x=6x(2x^2-3x+1).

B. Factoring Trinomials ax^2+bx+c

i. Simple trinomials (a = 1)

Factor x^2+7x+10.

Find two numbers that multiply to 10 and add to 7 → 5 and 2.

So, (x+5)(x+2).

ii. Advanced trinomials (a ≠ 1) – “AC” method

Factor 6x^2+11x+4.

Step 1: Multiply A and C (A is the leading coefficient and C is the constant)

6\cdot4=24

Step 2: Find two numbers that multiply to 24 and add to 11 → 8 and 3.

Step 3: Factor by grouping.

Step	Reason
6x^2 + 11x + 4	Original expression
6x^2 + 8x + 3x + 4	Rewrite the middle term using 8 and 3
(6x^2 + 8x) + (3x + 4)	Group terms
2x(3x + 4) + 1(3x + 4)	Factor each group
(3x + 4)(2x + 1)	Final factored form ✅

C. Special Patterns

• Difference of squares: a^2-b^2=(a+b)(a-b)
• Sum of cubes: a^3+b^3=(a+b)(a^2-ab+b^2)
• Difference of cubes: a^3-b^3=(a-b)(a^2+ab+b^2)

Example

Factor x^4-81.

First, notice x^4=(x^2)^2 and 81=9^2; therefore it is a difference of squares.

x^4-81=(x^2+9)(x^2-9).

However, x^2-9 is again a difference of squares:

x^2-9=(x+3)(x-3).

Final answer: (x^2+9)(x+3)(x-3).

D. Factoring Speeds Up Solving Quadratic Equations

Solve x^2-5x+6=0 by factoring.

• Factor: x-2(x-3)=0
• Zero Product Rule gives x=2 or x=3.
• Therefore, the quadratic is solved without a calculator or quadratic formula.

Rewriting Polynomials & Rational Expressions

A. Radicals and Rational Exponents

Remember, \sqrt[n]{x}=x^{1/n}. Therefore, converting forms can simplify multiplication.

Example

Rewrite \sqrt[3]{x^5} in rational-exponent form:

x^{5/3}.

B. Simplifying Rational Expressions by Factoring

Example

Simplify \dfrac{6x^2-15x}{3x}.

Step 1: Factor the numerator: 3x(2x-5).

Step 2: Cancel the common factor 3x with denominator 3x.

Result: 2x-5.

Therefore, recognizing common factors keeps the work short.

Strategy Section: Choosing the Right Tool on Test Day

Certain SAT® phrases signal the required operation.

• “Equivalent form that reveals zeros” → factor.
• “Expanded form” or “standard form” → multiply.

Moreover, watch for perfect-square or difference-of-squares patterns; spotting them avoids heavy algebra. Finally, keep mental math sharp by memorizing squares up to 15^2 and cubes up to 5^3.

Quick Reference Chart: Must-Know Vocabulary

Word	Definition	SAT® Tip
Polynomial	Sum of terms with non-negative integer exponents	Ensure no variables in denominators or radicals
Degree	Highest exponent in the polynomial	Shows the maximum number of roots
Coefficient	Number multiplied by a variable	Zero coefficient means term is absent
GCF	Greatest common factor of all terms	Factor it out first to simplify
Factor	Expression multiplied with another to produce a product	Used to reveal zeros quickly
Expand	Rewrite as a sum instead of a product	Often done by FOIL or distribution
Rational Expression	Fraction whose numerator and denominator are polynomials	Always factor first when simplifying
Radical	Expression containing a root symbol	Convert to exponents for easier manipulation
Exponent	Indicates repeated multiplication	Follow power rules for simplification
Like Terms	Terms with identical variable parts	Only these can be added or subtracted

Practice Question Set

Try these three mini SAT®-style problems.

1. Expand and simplify: (2x-1)(x+4)
2. Factor completely: 8y^2-18y+9
3. Simplify: \dfrac{x^2-16}{x+4}

Solutions

1. 2x^2+7x-4
2. (2y-3)(4y-3)
3. x-4

Final Takeaways & Next Steps

Mastering Operations with Polynomials transforms many SAT® questions into quick wins. Therefore, practice each skill until spotting patterns feels natural. Next, set a timer for 10 minutes and drill five factoring problems daily. For more targeted practice, explore SAT® Math study guides and full-length tests that focus on polynomial operations.

Unlock those points—one factor at a time!

Sharpen Your Skills for SAT® Math (Digital)

Are you preparing for the SAT® Math (Digital) test? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world SAT® Math (Digital) problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!

Need help preparing for your SAT® Math (Digital) exam?

Albert has hundreds of SAT® Math (Digital) practice questions, free response, and full-length practice tests to try out.