Polynomials: Basic Formulae of Algebra

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.

Degree of a Polynomial

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.

Division of a Polynomial by a Polynomial

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).

Factor Theorem

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.

Remainder Theorem

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).

Some Useful Formulas

  1. (a + b)2 = a2 + b2 + 2ab
  2. (a - b)2 = a2 + b2 - 2ab
  3. (a + b)2 - (a - b)2 = 4ab
  4. (a + b)2 + (a - b)2 = 2(a2 + b2)
  5. (a2 - b2) = (a + b)(a - b)
  6. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
  7. (a3 + b3) = (a +b)(a2 - ab + b2)       
  8. (a3 - b3) = (a - b)(a2 + ab + b2)
  9. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
  10. If a + b + c = 0, then a3 + b3 + c3 = 3abc