A function p(x) of the form p(x) = a0 + a1x + a2x2 + ... + anxn where a0, a1, a2, ..., an are real numbers, an ≠ 0 and n is a non-negative integer is called a polynomial in x.
The real number a0, a1, ..., an are called the coefficients of the polynomial.
Monomial: A polynomial having only one term is called a monomial. For example, 7, 2x, 8x3 are monomials.
Binomial: A polynomial having two terms is called a binomial. For example, 2x + 3, 7x2 - 4x, x2 + 8 are binomials.
Trinomial: A polynomial having three terms is called a trinomial. For example, 7x2 - 3x + 8 is a trinomial.
The exponent in the term with the highest power is called the degree of the polynomial.
For example, in the polynomial 8x6 - 4x5 + 7x3 - 8x2 + 3, the term with the highest power is x6. Hence, the degree of the polynomial is 6.
A polynomial of degree 1 is called a linear polynomial. It is of the form ax + b, a ≠ 0.
A polynomial of degree 2 is called a quadratic polynomial. It is of the form ax2 + bx + c, a ≠ 0.
Let p(x) and f(x) be two polynomials and f(x) ≠ 0. Then, if you can find polynomials q(x) and r(x), such that p(x) = f(x) . q(x) + r(x), where degree r(x) < degree f(x), then we say that p(x) divided by f(x), gives q(x) as quotient and r(x) as remainder.
If the remainder r(x) is zero, then the divisor f(x) is a factor of p(x) and p(x) = f(x) . q(x).
Let p(x) be a polynomial of degree n > 0. If p(a) = 0 for a real number a, then (x - a) is a factor of p(x).
Conversely, if (x - a) is a factor of p(x), then p(a) = 0.
Let p(x) be any polynomial of degree ≥ 1 and a any number. If p(x) is divided by x - a, the remainder is p(a).