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Friday, May 23, 2014

DEFINITIONS FROM REAL NUMBERS AND POLYNOMIALS (STANDARD X)



DEFINITIONS FROM REAL NUMBERS 

Real Numbers - The collection consisting of all rational and irrational numbers.

Euclid's Division Lemma or Euclid's Division Algorithm - For any two integers a and b there exist unique whole numbers q and r such that a = bq + r, where 0 is equal to or less than r < b.

Natural Numbers - Counting numbers such as 1,2,3,4......etc. are known as natural numbers.(excluding zero.)

Whole Numbers - All counting numbers together with 0 form the collection of whole numbers.

Integers - All counting numbers along with their negatives and zero form the collection of all integers.

Rational Numbers In Decimal Form - Every Rational Number when expressed in decimal form is expressed either in terminating or in non-terminating repeating decimal form.

Irrational Numbers - The numbers which when expressed in decimal form are expressible as 
non-terminating and non-repeating decimals.



                DEFINITIONS FROM POLYNOMIALS

Polynomials - An expression of the form p(x) = a(subscript 0) + a(subscript 1)x^1 + a(subscript 2)x^2 + a(subscript 3)x^3 + a (subscript 4)x^4......+a(subscript n)x^n is a polynomial in x of degree n.Here, a(subscript 0),a(subscript 1),a(subscript 2),a(subscript 3),a (subscript 4) are real numbers and each power of x is a non-negative integer.

Linear Polynomials - Polynomials of degree 1 are called Linear Polynomials.The standard form of a linear polynomial is p(x) = ax + b,where a is unequal to zero.

Quadratic Polynomials - Polynomials of degree 2 are called Quadratic Polynomials.The standard form of a quadratic polynomial is p(x) = ax^2 + bx +c,where a is unequal to zero.

Cubic Polynomials - Polynomials of degree 3 are called Cubic Polynomials.The standard forms of a cubic polynomial is p(x) = ax^3 + bx^2 +cx +d,where a is unequal to zero.

Biquadratic Polynomial - Polynomials of degree 4 are called Biquadratic Polynomials.The standard form of a biquadratic polynomial is p(x) = ax^4 + bx^3 +cx^2 +dx+e,where a is unequal to zero.

Value of a Polynomial At a Given Point - If p(x) is a polynomial and α is a real number,then the value obtained by putting x = α in p(x) is called the value of p(x) at x = α.

Zero Of A Polynomial - A real number α is called the zero of the polynomial p(x),if p(α) = 0.

Zero Polynomial - The constant polynomial P(x)=0 whose coefficients are all equal to 0.(Credit : www.mathworld.wolfram.com)

Division Algorithm For Polynomials - If f(x) and g(x) are any two polynomials with g(x) is unequal to zero,then we can find polynomials q(x) and r(x) such that f(x) = q(x)g(x) + r(x) where r(x) = 0 or {degree of r(x)} < {degree of g(x)}.

Credits : Secondary School Mathematics Class X Book,Bharati Bhawan (P & D) by R.S Aggarwal and V Aggarwal,unless any other credit is mentioned along with the definition.

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