SAT Math – Advanced Math
Operations with Polynomials
Adding, subtracting, multiplying, and dividing polynomial expressions
Polynomial operations are the fundamental algebraic manipulations that extend arithmetic to expressions with variables. On the SAT, you'll add, subtract, multiply, and divide polynomials, apply special product formulas, and simplify complex expressions—skills essential for advanced algebra, calculus, and mathematical modeling.
Success requires understanding like terms, applying the distributive property systematically, recognizing special product patterns (difference of squares, perfect square trinomials), and organizing polynomial division. These operations aren't just algebraic exercises—they're the foundation for analyzing functions, solving equations, and modeling everything from projectile motion to economic relationships.
Understanding Polynomial Operations
What is a Polynomial?
A polynomial is an expression with variables raised to whole number exponents, combined with coefficients.
Terms: Parts separated by + or - signs
Coefficients: Numbers multiplying variables (3, -5, 7)
Degree: Highest exponent (2 in this example)
Like Terms
Like terms have the same variables with the same exponents.
Unlike terms: \(3x^2\) and \(7x\) cannot be combined
Rule: Only like terms can be added/subtracted
Combine: Add or subtract coefficients, keep variable part
Special Products
Recognize patterns for faster multiplication.
Perfect square: \((a+b)^2 = a^2 + 2ab + b^2\)
Perfect square: \((a-b)^2 = a^2 - 2ab + b^2\)
Sum/diff of cubes: Special formulas (less common on SAT)
Distributive Property
The foundation of polynomial multiplication.
FOIL: \((a+b)(c+d) = ac + ad + bc + bd\)
General: Multiply each term in first by each term in second
Combine: Simplify by combining like terms
Essential Formulas and Rules
Adding and Subtracting Polynomials
1. Remove parentheses (distribute negative sign if subtracting)
2. Identify like terms
3. Combine coefficients of like terms
4. Write result in standard form (descending exponents)
FOIL Method (Binomial × Binomial)
\((a + b)(c + d) = ac + ad + bc + bd\)
First: Multiply first terms
Outer: Multiply outer terms
Inner: Multiply inner terms
Last: Multiply last terms
Special Product Formulas
\((a + b)^2 = a^2 + 2ab + b^2\)
\((a - b)^2 = a^2 - 2ab + b^2\)
\((a + b)(a - b) = a^2 - b^2\)
Polynomial Long Division
1. Divide leading term of dividend by leading term of divisor
2. Multiply entire divisor by quotient term
3. Subtract from dividend
4. Repeat with new dividend until degree is less than divisor
Common Pitfalls & Expert Tips
❌ Sign errors when distributing negatives
When subtracting, distribute the negative to ALL terms: \((3x - 5) - (2x + 4) = 3x - 5 - 2x - 4\), not \(3x - 5 - 2x + 4\)!
❌ Forgetting middle term in perfect squares
\((x + 3)^2 = x^2 + 6x + 9\), NOT \(x^2 + 9\). The middle term \(2ab\) is crucial!
❌ Combining unlike terms
\(3x^2 + 5x\) cannot be combined into \(8x^2\) or \(8x\). Different exponents = unlike terms!
❌ Missing terms in multiplication
When multiplying polynomials, must multiply EVERY term in first by EVERY term in second. Don't skip combinations!
✓ Expert Tip: Organize vertically for complex problems
When adding/subtracting, align like terms vertically. This prevents errors and makes combining terms systematic.
✓ Expert Tip: Check by substitution
Verify multiplication by substituting a simple value like x = 2. Both original and expanded form should give same result.
✓ Expert Tip: Recognize special products instantly
\((x + 5)(x - 5)\) immediately becomes \(x^2 - 25\). No FOIL needed when you spot the pattern!
Fully Worked SAT-Style Examples
Simplify: \((3x^2 + 5x - 7) + (2x^2 - 3x + 4)\)
Solution:
Step 1: Remove parentheses
\(3x^2 + 5x - 7 + 2x^2 - 3x + 4\)
Step 2: Group like terms
\((3x^2 + 2x^2) + (5x - 3x) + (-7 + 4)\)
Step 3: Combine coefficients
\(5x^2 + 2x - 3\)
Answer: \(5x^2 + 2x - 3\)
Simplify: \((4x^2 - 6x + 9) - (2x^2 + 3x - 5)\)
Solution:
Step 1: Distribute negative sign to all terms in second polynomial
\(4x^2 - 6x + 9 - 2x^2 - 3x + 5\)
Step 2: Group like terms
\((4x^2 - 2x^2) + (-6x - 3x) + (9 + 5)\)
Step 3: Combine
\(2x^2 - 9x + 14\)
Key Point:
Negative sign changes ALL signs in second polynomial
+3x becomes -3x, -5 becomes +5
Answer: \(2x^2 - 9x + 14\)
Multiply: \((2x + 3)(x - 5)\)
Solution:
Apply FOIL:
First: \(2x \cdot x = 2x^2\)
Outer: \(2x \cdot (-5) = -10x\)
Inner: \(3 \cdot x = 3x\)
Last: \(3 \cdot (-5) = -15\)
Combine all terms:
\(2x^2 - 10x + 3x - 15\)
Combine like terms:
\(2x^2 - 7x - 15\)
Answer: \(2x^2 - 7x - 15\)
Multiply: \((3x + 7)(3x - 7)\)
Solution:
Recognize pattern:
Form: \((a + b)(a - b)\)
This is difference of squares pattern!
Apply formula:
\((a + b)(a - b) = a^2 - b^2\)
Here: \(a = 3x\), \(b = 7\)
\((3x)^2 - 7^2 = 9x^2 - 49\)
Verify with FOIL:
\(9x^2 - 21x + 21x - 49 = 9x^2 - 49\) ✓
Middle terms cancel! Pattern recognition saves time.
Answer: \(9x^2 - 49\)
Expand: \((2x - 5)^2\)
Solution:
Recognize pattern:
\((a - b)^2 = a^2 - 2ab + b^2\)
Here: \(a = 2x\), \(b = 5\)
Apply formula:
\(a^2 = (2x)^2 = 4x^2\)
\(-2ab = -2(2x)(5) = -20x\)
\(b^2 = 5^2 = 25\)
Combine:
\(4x^2 - 20x + 25\)
Answer: \(4x^2 - 20x + 25\)
Multiply: \(3x(2x^2 - 5x + 4)\)
Solution:
Distribute 3x to each term:
\(3x \cdot 2x^2 = 6x^3\)
\(3x \cdot (-5x) = -15x^2\)
\(3x \cdot 4 = 12x\)
Combine all products:
\(6x^3 - 15x^2 + 12x\)
Answer: \(6x^3 - 15x^2 + 12x\)
Multiply: \((x + 2)(x - 3)(x + 1)\)
Solution:
Step 1: Multiply first two binomials
\((x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6\)
Step 2: Multiply result by third binomial
\((x^2 - x - 6)(x + 1)\)
Distribute each term:
\(x^2 \cdot x + x^2 \cdot 1 - x \cdot x - x \cdot 1 - 6 \cdot x - 6 \cdot 1\)
\(= x^3 + x^2 - x^2 - x - 6x - 6\)
Step 3: Combine like terms
\(x^3 + 0x^2 - 7x - 6 = x^3 - 7x - 6\)
Answer: \(x^3 - 7x - 6\)
Simplify: \(2(x + 3)^2 - (x - 1)(x + 4)\)
Solution:
Step 1: Expand \((x + 3)^2\)
\((x + 3)^2 = x^2 + 6x + 9\)
\(2(x^2 + 6x + 9) = 2x^2 + 12x + 18\)
Step 2: Expand \((x - 1)(x + 4)\)
\(x^2 + 4x - x - 4 = x^2 + 3x - 4\)
Step 3: Subtract (distribute negative)
\(2x^2 + 12x + 18 - (x^2 + 3x - 4)\)
\(= 2x^2 + 12x + 18 - x^2 - 3x + 4\)
Step 4: Combine like terms
\(x^2 + 9x + 22\)
Answer: \(x^2 + 9x + 22\)
Special Products Quick Reference
\((a + b)^2 = a^2 + 2ab + b^2\)
\((a - b)^2 = a^2 - 2ab + b^2\)
\((a + b)(a - b) = a^2 - b^2\)
Polynomial Operations: Building Algebraic Fluency
Mastering polynomial operations transforms algebra from memorized procedures into fluent mathematical thinking. Every time you add like terms, distribute correctly, or recognize a special product pattern, you're developing the algebraic intuition that distinguishes strong mathematics students. The SAT tests these skills because they're foundational for all higher mathematics—calculus begins with polynomial differentiation, linear algebra manipulates polynomial equations, and advanced functions are often approximated by polynomials. Beyond academics, polynomial operations model real-world relationships: the area of combined regions (adding polynomials), revenue functions (multiplying price by quantity polynomials), profit margins (subtracting cost from revenue), and countless optimization problems. Master the systematic approach: organize your work vertically, distribute carefully maintaining sign integrity, combine like terms methodically, and verify results. Recognize special products instantly to save time—when you see \((x+5)^2\), your brain should immediately produce \(x^2 + 10x + 25\) without FOIL. This fluency, built through practice and pattern recognition, is what enables you to focus on problem-solving rather than mechanical calculations when facing complex SAT questions or advanced mathematical challenges.