Polynomial Operations- Techniques and Examples
What Polynomials Actually Are
A polynomial is a mathematical expression with multiple terms connected by addition or subtraction. Each term contains a coefficient (the number in front) and variables raised to powers.
Examples:
- 3x² + 2x - 5
- 4a³ - 7a² + a + 9
- 2xy + 3y² - x + 4
The degree of a polynomial is the highest exponent. 3x⁴ + 2x² - 7 is a fourth-degree polynomial.
Adding Polynomials
Combine like terms. Like terms have the same variables raised to the same powers.
Example
(3x² + 2x - 5) + (4x² - 3x + 1)
Step 1: Group like terms
3x² + 4x² = 7x²
2x - 3x = -x
-5 + 1 = -4
Answer: 7x² - x - 4
Subtracting Polynomials
Distribute the negative sign first, then combine like terms. This trips up most people.
Example
(5x³ + 3x² - 2) - (2x³ - 4x² + 6)
Step 1: Distribute the negative
5x³ + 3x² - 2 - 2x³ + 4x² - 6
Step 2: Combine
5x³ - 2x³ = 3x³
3x² + 4x² = 7x²
-2 - 6 = -8
Answer: 3x³ + 7x² - 8
Multiplying Polynomials
This is where most algebra problems come from. You need to master the distributive property and FOIL.
Multiplying a Monomial by a Polynomial
Multiply the monomial by every term in the polynomial.
Example
3x²(2x³ + 4x - 5)
3x² × 2x³ = 6x⁵
3x² × 4x = 12x³
3x² × (-5) = -15x²
Answer: 6x⁵ + 12x³ - 15x²
Multiplying Two Binomials — FOIL
FOIL stands for First, Outer, Inner, Last. Apply it every time you multiply two binomials.
Example
(x + 3)(x + 5)
First: x × x = x²
Outer: x × 5 = 5x
Inner: 3 × x = 3x
Last: 3 × 5 = 15
Combine: x² + 5x + 3x + 15 = x² + 8x + 15
Multiplying Any Two Polynomials
Use the distributive property repeatedly. Multiply every term in the first polynomial by every term in the second.
Example
(x + 2)(x² + 3x - 4)
x(x²) = x³
x(3x) = 3x²
x(-4) = -4x
2(x²) = 2x²
2(3x) = 6x
2(-4) = -8
Combine: x³ + 3x² + 2x² - 4x + 6x - 8
= x³ + 5x² + 2x - 8
Special Products You Should Memorize
These patterns appear constantly. Save time by recognizing them instantly.
| Pattern | Formula | Example |
|---|---|---|
| Difference of Squares | (a + b)(a - b) = a² - b² | (x + 4)(x - 4) = x² - 16 |
| Perfect Square Trinomial | (a + b)² = a² + 2ab + b² | (x + 3)² = x² + 6x + 9 |
| Perfect Square Trinomial | (a - b)² = a² - 2ab + b² | (x - 5)² = x² - 10x + 25 |
Dividing Polynomials
Two methods exist. Use polynomial long division for most cases, synthetic division for divisors of the form (x - c).
Polynomial Long Division
Works like regular long division. Divide, multiply, subtract, bring down, repeat.
Example
Divide x² + 5x + 6 by (x + 2)
Step 1: Divide first term: x² ÷ x = x
Step 2: Multiply: x(x + 2) = x² + 2x
Step 3: Subtract: (x² + 5x) - (x² + 2x) = 3x
Step 4: Bring down +6: now 3x + 6
Step 5: Divide: 3x ÷ x = 3
Step 6: Multiply: 3(x + 2) = 3x + 6
Step 7: Subtract: (3x + 6) - (3x + 6) = 0
Answer: x + 3
Synthetic Division
Faster method, but only works when dividing by (x - c). Use the opposite sign of the constant.
Example
Divide x² + 5x + 6 by (x + 2) using synthetic division
Write coefficients: 1, 5, 6
Use -2 (opposite of +2):
| -2 | | 1 | 5 | 6 |
| -2 | -6 | ||
| 1 | 3 | 0 |
Bring down 1, multiply -2 × 1 = -2, add 5 + (-2) = 3, multiply -2 × 3 = -6, add 6 + (-6) = 0
Answer: x + 3 ✓
Factoring Polynomials
Factoring is the reverse of multiplication. You're breaking a polynomial into simpler pieces that multiply together.
Factoring Out the GCF
Find the greatest common factor of all terms and factor it out.
Example
6x³ + 9x² - 3x
GCF of coefficients: 3
GCF of variables: x
GCF = 3x
Factor: 3x(2x² + 3x - 1)
Factoring Trinomials (x² + bx + c)
Find two numbers that multiply to c and add to b.
Example
x² + 7x + 12
Find two numbers that multiply to 12 and add to 7: 3 and 4
Rewrite: x² + 3x + 4x + 12
Group: (x² + 3x) + (4x + 12)
Factor: x(x + 3) + 4(x + 3)
= (x + 3)(x + 4)
Difference of Squares
a² - b² = (a + b)(a - b)
Example
16x² - 25
= (4x)² - 5²
= (4x + 5)(4x - 5)
Quick Reference: Operations Summary
| Operation | Key Step | Watch Out For |
|---|---|---|
| Addition | Combine like terms | Don't add unlike terms |
| Subtraction | Distribute negative first | Sign errors |
| Multiplication | FOIL for binomials, distribute for larger | Missing terms |
| Division | Long division or synthetic | Synthetic only works for (x - c) |
| Factoring | Find GCF first, then look for patterns | Forgetting to check GCF |
Getting Started: Practice Framework
Work through problems in this order:
- Identify the operation — addition, subtraction, multiplication, division, or factoring
- Check for a GCF — always factor it out first
- Apply the appropriate method — use the tables above to match the situation
- Combine like terms — simplify the final answer
- Verify by checking the result — multiply back to confirm
Most mistakes happen when students skip step 1 or forget step 4. The problems aren't hard — the errors are usually simple sign mistakes or forgetting to combine like terms at the end.