Operations with Polynomials Guide
What Are Polynomials?
A polynomial is an expression with multiple terms made up of constants, variables, and exponents. The exponents must be whole numbers—0, 1, 2, 3, and so on. If you see a variable raised to a negative power or a fractional exponent, that's not a polynomial.
Polynomials come in different sizes:
- Monomial — one term (like 5x²)
- Binomial — two terms (like x + 3)
- Trinomial — three terms (like x² + 5x + 6)
- Anything with more terms is just called a polynomial
The degree of a polynomial is the highest exponent. x⁴ + 2x² - 7 is a 4th-degree polynomial. This matters because degree affects how you work with them.
Adding Polynomials
You can only combine like terms—terms with the same variable raised to the same power. That's it. Don't try to add x² and x together. They're different.
Example: (3x² + 2x + 5) + (4x² - 3x + 1)
Add the coefficients of like terms:
- 3x² + 4x² = 7x²
- 2x + (-3x) = -x
- 5 + 1 = 6
Result: 7x² - x + 6
When polynomials are stacked vertically, align like terms in columns and add down. This prevents mistakes with larger expressions.
Subtracting Polynomials
Subtraction trips people up. You must distribute the negative sign to every term in the polynomial being subtracted.
Example: (5x³ + 2x² - 4x) - (3x³ - 6x² + x)
Distribute the minus sign:
5x³ + 2x² - 4x - 3x³ + 6x² - x
Now combine like terms:
- 5x³ - 3x³ = 2x³
- 2x² + 6x² = 8x²
- -4x - x = -5x
Result: 2x³ + 8x² - 5x
Forgetting to distribute the negative is the #1 mistake here. Write it out explicitly if you have to.
Multiplying Polynomials
Multiplication requires the distributive property. Every term in the first polynomial gets multiplied by every term in the second polynomial.
Monomial × Polynomial
Multiply the monomial by each term individually.
Example: 3x²(2x³ + 4x - 5)
- 3x² × 2x³ = 6x⁵
- 3x² × 4x = 12x³
- 3x² × (-5) = -15x²
Result: 6x⁵ + 12x³ - 15x²
Binomial × Binomial (FOIL)
FOIL works for multiplying two binomials. It stands for First, Outer, Inner, Last—the four pairs of terms you multiply together.
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
FOIL is just a memory device. The real rule is: multiply everything by everything.
Multiplying Larger Polynomials
For (x + 2)(x² + 3x - 4), distribute each term from the first polynomial:
- x(x² + 3x - 4) = x³ + 3x² - 4x
- 2(x² + 3x - 4) = 2x² + 6x - 8
Add the results: x³ + 3x² - 4x + 2x² + 6x - 8 = x³ + 5x² + 2x - 8
Special Products
Two patterns come up constantly. Memorize them.
Difference of squares: (a + b)(a - b) = a² - b²
Example: (x + 4)(x - 4) = x² - 16
Perfect square trinomials:
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
Example: (x + 3)² = x² + 6x + 9
These shortcuts save time on tests. Verify by expanding if you're unsure.
Dividing Polynomials
Dividing by a Monomial
Split the numerator into separate fractions, one for each term.
Example: (12x⁴ + 8x² - 4x) ÷ 4x
Write as fractions:
(12x⁴/4x) + (8x²/4x) - (4x/4x)
Simplify each:
- 12x⁴ ÷ 4x = 3x³
- 8x² ÷ 4x = 2x
- 4x ÷ 4x = 1
Result: 3x³ + 2x + 1
Polynomial Long Division
For dividing by polynomials with more than one term, use long division. It's the same process as dividing numbers, but with variables.
Example: (x² + 7x + 12) ÷ (x + 3)
Step 1: Divide the first term of the dividend by the first term of the divisor.
x² ÷ x = x
Step 2: Multiply the entire divisor by this result and subtract.
x(x + 3) = x² + 3x
Subtract: (x² + 7x + 12) - (x² + 3x) = 4x + 12
Step 3: Repeat. Divide 4x by x = 4.
Multiply: 4(x + 3) = 4x + 12
Subtract: (4x + 12) - (4x + 12) = 0
Result: x + 4
If you get a remainder, write it as a fraction over the divisor. Example: quotient + remainder/(divisor).
Quick Reference: Operations Summary
| Operation | Rule | Key Point |
|---|---|---|
| Addition | Combine like terms | Only add terms with same variable and exponent |
| Subtraction | Distribute negative, then combine | Subtract every term, not just the first |
| Multiplication | Distribute to all terms | Every term × every term |
| Division by monomial | Split into separate fractions | Divide each term individually |
| Division by polynomial | Long division | Repeat until degree of remainder < degree of divisor |
How to Practice: Getting Started
Work through these steps to build solid polynomial skills:
- Start with addition and subtraction — focus on identifying like terms correctly. Write out terms with their signs explicitly.
- Move to multiplication — begin with monomial × polynomial, then binomials. Use FOIL for binomials until it's automatic.
- Learn the special products — difference of squares and perfect square trinomials. These appear constantly in factoring and solving equations.
- Tackle division — monomial division first, then long division. Check your work by multiplying the quotient by the divisor.
- Practice with missing terms — polynomials like x³ + 5 have missing x² and x terms (coefficient 0). Include placeholders to avoid alignment errors in vertical operations.
Always verify answers by substituting a value for x. If (x + 2)(x + 3) = x² + 5x + 6, test x = 1: (1 + 2)(1 + 3) = 3 × 4 = 12, and 1² + 5(1) + 6 = 12. It works.
Polynomial operations are mechanical once you understand the rules. The only way to get faster is repetition. Work problems daily until the steps become reflex.