Polynomials – Expressions of Power
A polynomial is an expression made up of one or more terms, each consisting of a variable raised to a non-negative whole-number power. Polynomials are one of the most fundamental structures in mathematics.
What Is a Polynomial?
A polynomial is a sum of terms of the form ax to the power n, where a is a coefficient and n is a non-negative integer. Examples: 3x squared + 2x - 5 and x cubed - 4. NOT polynomials: 1/x (negative power) or the square root of x (fractional power).
Naming Polynomials by Degree
| Highest Power | Name | Example |
|---|---|---|
| 0 | Constant | 7 |
| 1 | Linear | 3x + 2 |
| 2 | Quadratic | x squared - 4x + 1 |
| 3 | Cubic | 2x cubed + x - 3 |
| 4 | Quartic | x to the 4 - 5x squared |
Adding and Subtracting Polynomials
Collect like terms — match powers and combine their coefficients.
x squared terms: 3 + 1 = 4. x terms: 2 + (-5) = -3. Constants: -1 + 4 = 3. Answer: 4x squared - 3x + 3.
(5x squared + 4x - 1) - (2x squared - x + 3) = 5x squared - 2x squared + 4x + x - 1 - 3 = 3x squared + 5x - 4.
Multiplying Polynomials
Multiply each term in the first polynomial by each term in the second, then collect like terms.
x times x squared = x cubed. x times -3x = -3x squared. x times 1 = x. 2 times x squared = 2x squared. 2 times -3x = -6x. 2 times 1 = 2. Collect: x cubed + (-3+2)x squared + (1-6)x + 2 = x cubed - x squared - 5x + 2.
The Remainder Theorem
When a polynomial P(x) is divided by (x - a), the remainder equals P(a). This saves performing full polynomial long division.
P(2) = 2(8) - 3(2) + 1 = 16 - 6 + 1 = 11. Remainder is 11.
Key Takeaways
- A polynomial has non-negative integer powers of variables only.
- The degree is the highest power; it names the polynomial (linear, quadratic, cubic, etc.).
- Add/subtract by collecting like terms; multiply by distributing each term.
- The Remainder Theorem: when P(x) is divided by (x-a), the remainder is P(a).
Practice Questions
- Add (4x cubed - 2x + 5) and (x cubed + 3x squared - 4x - 2).
- Subtract (3x squared - x + 2) from (7x squared + 4x - 5).
- Expand (x + 3)(x squared - x + 2).
- What is the degree of 5x cubed - 2x squared + x - 7?
- Find the remainder when P(x) = x cubed - 4x + 6 is divided by (x - 1).