A polynomial is a mathematical expression consisting of variables (also known as indeterminates) and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. The general form of a polynomial is:
P(x) = anxn + an-1xn-1 + ....... + a2x2 + a1x1 + a0
- P(x) represents the polynomial.
- an, an-1, … , a2, a1, a0 are coefficients, and anan is the leading coefficient.
- n is a non-negative integer, and it represents the degree of the polynomial.
- The terms anxn, an-1xn-1, ....... , a2x2, a1x1, a0 are called the monomials, and the sum of these monomials forms the polynomial.
Here are a few examples of polynomials:
- P(x) = 3x2−5x+2
(a quadratic polynomial with degree 2)
- Q(y) = 4y3 + 2y2 − 7y + 1
(a cubic polynomial with degree 3)
- R(t) = 6t4 − t2 + 5
(a quartic polynomial with degree 4)
- S(z) = 2z + 1
(a linear polynomial with degree 1)
- T(w) = 8
(a constant polynomial with degree 0)
The degree of a polynomial is determined by the highest exponent of the variable in the expression. For example, in P(x) = 3x2 − 5x + 2 the degree is 2 because the highest exponent of x is 2.
Polynomials are fundamental in algebra and mathematics, and they are used in various areas such as solving equations, curve fitting, and approximation.
The study of polynomials involves understanding their properties, operations (addition, subtraction, multiplication), and factorization.
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