Polynomial Operations
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn how to add, subtract, and multiply polynomials, including how to organize terms and simplify the results.
The Concept
Section titled “The Concept”Polynomial operations follow the same rules as working with numbers, but you must combine like terms (terms with the same variable and same exponent).
Addition and Subtraction - combine like terms. When subtracting, distribute the negative sign to every term in the second polynomial.
Multiplication - use the distributive property. Multiply every term in the first polynomial by every term in the second. For two binomials, this is called FOIL (First, Outer, Inner, Last).
The diagram shows the FOIL method for multiplying two binomials. Each colored arc connects two terms being multiplied: blue for First (x · 2x = 2x²), green for Outer (x · −3 = −3x), orange for Inner (4 · 2x = 8x), and purple for Last (4 · −3 = −12). Combine the like terms (−3x + 8x = 5x) to get the final answer: 2x² + 5x − 12.
Important reminders:
- When subtracting, distribute the negative sign to every term in the second polynomial.
- Write the final answer in standard form (descending powers of x).
- The degree of the product equals the sum of the degrees of the factors.
Worked Example
Section titled “Worked Example”1. Add two polynomials
(3x³ − 2x² + 5x − 1) + (x³ + 4x² − 7x + 8)
2. Subtract two polynomials
(5x² + 3x − 4) − (2x² − 6x + 9)
Distribute the negative sign first:
3. Multiply a binomial by a trinomial
(x + 4)(2x² − 3x + 5)
Distribute each term of the first polynomial across the second:
Real-World Application
Section titled “Real-World Application”Polynomial operations are used when combining real quantities:
- Total cost: fixed costs + variable costs (often polynomials).
- Profit: revenue polynomial minus cost polynomial.
- Area/volume: multiplying length, width, and height expressions.
- Physics: combining force, velocity, or acceleration models.
- Economics: total revenue = price × quantity, where both may be polynomial expressions.
Example: a company’s revenue is r(x) = 50x − 2x² and cost is c(x) = 200 + 15x. Profit = r(x) − c(x) = 50x − 2x² − 200 − 15x = −2x² + 35x − 200.