What is a Polynomial?
The word polynomial comes from the Greek poly (many) and Latin nomen (name/term). A polynomial in x looks like:
aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
Where aₙ, …, a₀ are real-number coefficients and n is a non-negative integer.
Degree of a Polynomial
The degree is the highest exponent of the variable in the polynomial.
| Polynomial | Degree | Name |
|---|---|---|
| 7 | 0 | Constant |
| 3x + 1 | 1 | Linear |
| x² - 4x + 4 | 2 | Quadratic |
| 2x³ + x | 3 | Cubic |
| x⁴ - 1 | 4 | Quartic |
Standard Form
Standard form writes terms in descending order of degree:
4x³ - 2x² + x - 8 ✓ (standard form)
x - 2x² + 4x³ - 8 ✗ (not standard)
x - 2x² + 4x³ - 8 ✗ (not standard)
Adding & Subtracting Polynomials
Combine like terms (same degree):
(3x² + 2x - 1) + (x² - 5x + 4)
= (3+1)x² + (2-5)x + (-1+4)
= 4x² - 3x + 3
= (3+1)x² + (2-5)x + (-1+4)
= 4x² - 3x + 3
Multiplying Polynomials
Multiply every term in the first polynomial by every term in the second, then combine like terms:
(x + 2)(x² - 3x + 1)
= x³ - 3x² + x + 2x² - 6x + 2
= x³ - x² - 5x + 2
= x³ - 3x² + x + 2x² - 6x + 2
= x³ - x² - 5x + 2
Roots (Zeros) of a Polynomial
Roots are the values of x that make the polynomial equal to zero. A polynomial of degree n has at most n real roots. You can find them by factoring or using the quadratic formula for degree 2.