POLYNOMIALS

WHAT IS POLYNOMIALS ?

In mathematics, a polynomial is an expression consisting of variables (or indeterminate) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.

DEFINITION OF POLYNOMIALS

A polynomial in a single indeterminate can be written in the form

Generally elements of a ring, and x is a symbol which is called an indeterminate or, for historical reasons, a variable. The symbol x does not represent any value, although the usual (commutative, distributive) laws valid for arithmetic operations also apply to it.

Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms".

ARITHMETIC OF POLYNOMIALS

Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering, and combining of like terms. For example, if

P + Q= 3x^2 - 2x + 5xy - 2 -3x^2 + 3x + 4y^2 + 8

Then,

P + Q = 3x^2 - 2x + 5xy - 2 - 3x^2 + 3x + 4y^2 + 8

which can be simplified to-

P + Q = x + 5xy + 4y^2 + 6

POLYNOMIAL TERMS

Monomials :- an algebraic expression consisting of one term.

Binomial :- an algebraic expression of the sum or the difference of two terms.

Trinomial :- an algebraic expression of three terms.

Degree of polynomials :_ the highest power of the variable in polynomial is termed as the degree of polynomials.

Constant polynomials :- a polynomial of degree zero is called constant polynomials.

Linear polynomials :- a polynomial of degree one.

E.g. :- 9x+1

Quadratic polynomial :- a polynomial of degree two.

E.g. :- 3/2y^2 -3y +3

Cubic polynomial :- a polynomial of degree three.

E.g. :- 12x^3 -4x^2 +5x+1…

STANDARD FORMThe Standard Form for writing a polynomial is to put the terms with

the highest degree first.

Example: Put this in Standard form: x^2 – x^4 + x

The highest degree is 4, so that goes first, then x^2, x.

REMINDER THEOREM

• Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by linear polynomial x-a then the reminder if p(a).

• Proof :- let p(x) be any polynomial of degree greater than or equal to 1. Suppose that when p(x) is divided by x-a, then the quotient is q(x) and the reminder is r(x), e.g.:- p(x) +(x-a) q(x) +r(x)

Since the degree of x-a is 1 and the degree of r(x) is less than the degree of x-a, the degree of r(x) = 0.

This means that r(x) is a constant . Say r.

So , for every value of x, r(x) = r.

Therefore, p(x) = (x-a) q(x) + r

In particular , if x =a, this equation gives us

p(a)=(a-a) q(a) + r

Which proves the theorem.

IDENTITIES

If an equality holds true for all values of the variable, then its called identity.

NOTES:

A polynomial can have constants, variables and exponents, but never division by a variable

Degree :-The degree of a monomial is the sum of the exponents of its variables.

The degree of a polynomial is the highest degree of any of its terms, after it has been simplified.

By SHAFI…class 9