Polynomial Calculator
Perform polynomial operations including addition, subtraction, multiplication, and evaluation. Enter coefficients from highest to lowest degree.
What is a polynomial and how is it structured?
A polynomial is an expression with variables and coefficients using only addition, subtraction, multiplication, and non-negative integer exponents. Structure: aₙxⁿ + aₙ₋₁xⁿ⁻^1 + ... + a₁x + a₀, where aₙ are coefficients and n is the degree. Example: 3x⁴ - 2x^2 + 7x - 5 has degree 4. Terms are named: constant (x⁰), linear (x^1), quadratic (x^2), cubic (x^3), quartic (x⁴), quintic (x⁵).
How do you add and subtract polynomials?
Combine like terms (same degree). Example: (3x^2 + 2x - 1) + (x^2 - 5x + 4) = (3+1)x^2 + (2-5)x + (-1+4) = 4x^2 - 3x + 3. For subtraction, distribute the negative: (3x^2 + 2x - 1) - (x^2 - 5x + 4) = 3x^2 + 2x - 1 - x^2 + 5x - 4 = 2x^2 + 7x - 5. Align terms by degree and combine coefficients.
How do you multiply polynomials?
Use distributive property (FOIL for binomials). Every term multiplies every other term: (2x + 3)(x^2 - x + 1) = 2x(x^2) + 2x(-x) + 2x(1) + 3(x^2) + 3(-x) + 3(1) = 2x^3 - 2x^2 + 2x + 3x^2 - 3x + 3 = 2x^3 + x^2 - x + 3. Combine like terms. Degree of product = sum of degrees (linear * quadratic = cubic).
What does it mean to evaluate a polynomial?
Evaluating means finding the polynomial's value for a specific x. Substitute x and calculate: For P(x) = 2x^3 - 3x + 1, evaluate at x=2: P(2) = 2(2)^3 - 3(2) + 1 = 2(8) - 6 + 1 = 16 - 6 + 1 = 11. Use order of operations: exponents first, then multiplication, then addition/subtraction. Used for graphing, finding zeros, and applications.
What is the degree of a polynomial and why does it matter?
Degree is the highest exponent with non-zero coefficient. Determines: 1) Number of possible roots (up to degree n), 2) End behavior of graph, 3) Maximum turning points (n-1), 4) Complexity. Examples: 5x^3 - 2x + 1 (degree 3), x⁴ (degree 4), 7 (degree 0). Zero polynomial has no degree. Even degree: same end behavior both sides. Odd degree: opposite end behavior.
How do you write polynomials in standard form?
Standard form: terms ordered from highest to lowest degree with no like terms. Example: 3x + 5x^3 - 2 + x^2 becomes 5x^3 + x^2 + 3x - 2. Steps: 1) Combine like terms, 2) Arrange by decreasing degree, 3) Write missing degrees as 0 if needed (5x^3 + 0x^2 + 3x - 2). Coefficient of highest term (5) is the leading coefficient. Makes comparison and operations easier.
What are common applications of polynomials?
Real-world uses: 1) Physics - projectile motion (h = -16t^2 + v₀t + h₀), 2) Economics - cost/revenue functions, 3) Engineering - stress analysis, beam deflection, 4) Chemistry - reaction rates, 5) Computer graphics - curve modeling (Bezier curves), 6) Statistics - polynomial regression for trend analysis. Higher degrees model more complex relationships but risk overfitting.
What is the difference between monomials, binomials, and trinomials?
Classification by number of terms: Monomial = 1 term (3x^2, -5, 7x⁴). Binomial = 2 terms (x^2 + 5, 3x - 2). Trinomial = 3 terms (x^2 + 3x - 4, 2x^3 - x + 1). More terms = polynomial (general term). Special products: (a+b)^2 = a^2 + 2ab + b^2 (binomial squared gives trinomial). Knowing types helps identify factoring patterns and solution methods.
How do you find the zeros or roots of a polynomial?
Methods depend on degree: Linear (ax+b): x = -b/a. Quadratic (ax^2+bx+c): quadratic formula, factoring, completing square. Cubic/higher: factoring, synthetic division, rational root theorem, numerical methods (Newton's). Fundamental Theorem: degree n polynomial has exactly n roots (counting multiplicity, including complex). Example: x^2 - 5x + 6 = (x-2)(x-3) has roots x=2,3. Graphically: where graph crosses x-axis.
Can you divide polynomials and what methods exist?
Yes, via long division or synthetic division (simpler when dividing by linear factor). Process like number division: (x^3 + 2x^2 - 5x + 6) / (x - 1) = x^2 + 3x - 2 remainder 4. Result: Quotient + Remainder/Divisor. Remainder Theorem: P(x) / (x-a) has remainder P(a). Factor Theorem: (x-a) is factor if P(a)=0. Used for simplification, finding factors, partial fractions.