X 4 − x 2 + 1 is a non-monic polynomials of degree 4 with leading coefficient and constant term 1. Thus x 5 − 2 x 3 + 8 x + 3 is a monic polynomial of degree 5 with constant term 3, while The term independent of is called the constant term. The index of the leading term is called the degree of the polynomial. If the leading coefficient is 1, then the polynomial is called monic. The term with the highest power of is called the leading term and its coefficient is called the leading coefficicent. There may be any number of terms, but each term must be a multiple of a whole number power of x. One such application to error-correcting codes is discussed in the Appendix to this module.Ī polynomial is as expression such as x 5 − 2 x 3 + 8 x + 3 or x 4 − x 2 + 1. Similarly, information about the roots of a polynomial equation enables us to give a rough sketch of the corresponding polynomial function.Īs well as being intrinsically interesting objects, polynomials have important applications in the real world. The number of solutions of the this equation assisted us in drawing the graph of the quadratic function y = ax 2 + bx + c. Just as we can divide one whole number by another, producing a quotient and remainder, we can divide one polynomial by another and obtain a quotient and remainder, which are also polynomials.Ī quadratic equation of the form ax 2 + bx + c has either 0, 1 or 2 solutions, depending on whether the discriminant is negative, zero or positive. We can add, subtract and multiply two or more polynomials together to obtain another polynomial. Polynomials in many respects behave like whole numbers or the integers. In this module we will see how to arrive at this factorisation. Similarly we can factor the cubic x 3 − 6 x 2 + 11 x − 6 as ( x − 1)( x − 2)( x − 3), which enables us to show that the solutions of x 3 − 6 x 2 + 11 x − 6 = 0 are x = 1, x = 2 or x = 3. The quadratic x 2 − 5 x + 6 factors as ( x − 2)( x − 3). How these things might be performed with algebraic expressions of higher degree. We can factor quadratic expressions, solve quadratic equations and graph quadratic functions, the obvious question arises as to Indeed a quadratic is a polynomial of degree 2.

Polynomials represent the next level of algebraic complexity after quadratics.

Is 3x4 2x 5 even or odd