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  • Which of the following is a polynomial?Option: 1 <img alt="x^{2}-5 x+4 \sqrt{x}+4" src="https://entrancecorner.oncodecogs.com/gif.latex?x%5E%7B2%7D-5

Which of the following is a polynomial?

Option: 1

x^{2}-5 x+4 \sqrt{x}+4


Option: 2

x^{5 / 2}-x+x^{1 / 2}+1


Option: 3

\sqrt x+\frac{1}{\sqrt x}


Option: 4

\sqrt{2} x^{2}-3 \sqrt{3} x+\sqrt{3}


Answers (1)

 

 

Introduction -

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

A polynomial can have:

  • Constants like (2, -4, 10, ½ etc) 

  • Variables (like x, y and z)

  • Exponents (like 2 in x2), but only 0, 1, 2, 3, ... etc are allowed

A polynomial p(x) is an algebraic expression that can be written in the form of 

p(x)=a_{n} x^{n}+\ldots+a_{2} x^{2}+a_{1} x+a_{0}

Each real number ai is called a coefficient. The number a0  that is not multiplied by a variable is called a constant. Each product  a_{i}x_{i} is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called the degree of a polynomial. The leading term is the term with the highest power, and its coefficient is called the leading coefficient. 

Example

(i) 3 x+5 is a polynomial in x of degree 1

(ii) 8 x^{2}-5 x+3 is a polynomial in x of degree 2

(iii) 2 y^{3}+\frac{4}{9} y^{2}-5 y+\sqrt{3} is a polynomial in y of degree 3

(iv) 3 z^{4}-5 z^{3}+2 z^{2}-8 z+1 is a polynomial in z of degree 4

 

A polynomial in one variable:

The algebraic expressions that involve only one variable are called polynomials in one variable.

For example: 

8x, x^2+5, y^3+10  etc each involves one variable either x or y. 

Polynomials in two or more variables :

An algebraic expression, whose term or involves/involve two or more variables (literals) such that the exponent of each variable is a whole number, is called a polynomial in two or more variables.

For example:

3 x^{2}-6 x y+8 y^{2} is a polynomial in two variables x and y 

x+x y^{3}-8 x^{2} y z-15 is a polynomial in three variables x, y and z

-

x^{2}-5 x+4 \sqrt{x}+4 is not a polynomial 

x^{5 / 2}-x+x^{1 / 2}+1 is not a polynomial

\sqrt x+\frac{1}{\sqrt x} is not apolynomial

because degree of x is not whole number.

 

 

Posted by

Sumit Saini

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