Introduction to Polynomials
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn what polynomials are, the different types (monomial, binomial, trinomial), and how to add and subtract them.
The Concept
Section titled “The Concept”A polynomial is an expression with variables, coefficients, and exponents (whole numbers only), combined with + or −.
Examples:
- 5x² + 3x − 7 (trinomial)
- 4x³ (monomial)
- 2x + 5 (binomial)
- 9 (constant polynomial)
Terms: Each part separated by + or − (5x², 3x, −7 are terms).
Degree: Highest exponent (5x² + 3x − 7 has degree 2).
Leading coefficient: Coefficient of highest-degree term (5 in 5x² + …).
Adding/subtracting polynomials: Combine like terms (same variable and exponent).
Example: (3x² + 5x − 2) + (2x² − 4x + 7)
= (3x² + 2x²) + (5x − 4x) + (−2 + 7) = 5x² + x + 5
Subtract: (4x³ − 2x + 1) − (x³ + 3x² − 5)
= 4x³ − 2x + 1 − x³ − 3x² + 5 = 3x³ − 3x² − 2x + 6
Distribute the negative when subtracting: −(x³ + 3x² − 5) = −x³ − 3x² + 5
Worked Example
Section titled “Worked Example”Add: (6x² − 4x + 9) + (3x² + 5x − 2)
- Combine like terms: (6x² + 3x²) + (−4x + 5x) + (9 − 2)
- 9x² + x + 7
Subtract: (5x³ + 2x² − 8) − (2x³ − 3x + 1)
- Distribute negative: 5x³ + 2x² − 8 − 2x³ + 3x − 1
- (5x³ − 2x³) + 2x² + 3x + (−8 − 1) = 3x³ + 2x² + 3x − 9
Real-World Application
Section titled “Real-World Application”Polynomials model many real situations:
- Area of a rectangle: length × width = (x + 5)(x + 3) = x² + 8x + 15 (area in square units)
- Total cost: fixed cost + variable cost per item × number of items = 200 + 12x
- Profit: revenue − cost = (15x) − (200 + 8x) = 7x − 200 (x = units sold)
- Volume of a box: length × width × height = (x + 2)(x + 2)(x + 4) (expands to polynomial)
Adding/subtracting polynomials combines like terms in budgeting, area, profit, or volume calculations.