Question

In an algebraic expression , if the power of the variable are whole numbers then that algebraic expression is known as polynomial.When we perform a division of polynomial, we may get a remainder zero or non zero number. We can directly find the remainder while performing division of polynomial.Wehave studied new methods of division is std 9. Along with that we have also studied new concepts related with division. a) Which are the new methods of division that we studied in the chapter polynomial? b) Which are the two new theorems we studied and what is the difference between them? c) Suppose ‘x’ is the remainder of particular division. Is it possible that x < 0?​

Answer 1

Answer:

Polynomials represent the next level of algebraic complexity after quadratics. Indeed a quadratic is a polynomial of degree 2. We can factor quadratic expressions, solve quadratic equations and graph quadratic functions, the obvious question arises as to

how these things might be performed with algebraic expressions of higher degree.

The quadratic x2 − 5x + 6 factors as (x − 2)(x − 3). Hence the equation x2 − 5x + 6 = 0

has solutions x = 2 and x = 3.

Similarly we can factor the cubic x3 − 6x2 + 11x − 6 as (x − 1)(x − 2)(x − 3), which enables us to show that the solutions of x3 − 6x2 + 11x − 6 = 0 are x = 1, x = 2 or x = 3. In this module we will see how to arrive at this factorisation.

Polynomials in many respects behave like whole numbers or the integers. We can add, subtract and multiply two or more polynomials together to obtain another polynomial. 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.

A quadratic equation of the form ax2 + bx + c has either 0, 1 or 2 solutions, depending on whether the discriminant is negative, zero or positive. The number of solutions of the this equation assisted us in drawing the graph of the quadratic function y = ax2 + bx + c. Similarly, information about the roots of a polynomial equation enables us to give a rough sketch of the corresponding polynomial function.