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Question

Which of the following expressions are polynomials? In case of a polynomial, write its degree. 

(i) x52x3+x+3

(ii) y3+3y

(iii) t225t+5

(iv) x10001

(v) 12x22x+2

(vi) x2+2x1+3

(vii) 1

(viii) 35

(ix) x222x2

(x) 22x28

(xi) 12x2

(xii) 15x12+1

(xiii) 35x273x+9

(xiv) x4x32+x3

(xv) 2x3+3x2+x1
 


Solution

(i) x52x3+x+3 is an expression having only non-negative integral powers of x. So, it is a polynomial. Also, the highest power of x is 5, so, it is a polynomial of degree 5.

(ii) y3+3y is an expression having only non-negative integral powers of y. So, it is a polynomial. Also, the highest power of y is 3, so, it is a polynomial of degree 3.

(iii) t225t+5 is an expression having only non-negative integral powers of t. So, it is a polynomial. Also, the highest power of t is 2, so, it is a polynomial of degree 2.

(iv) x10001 is an expression having an only non-negative integral power of x. So, it is a polynomial. Also, the highest power of x is 100, so, it is a polynomial of degree 100.

(v) 12x22x+2 is an expression having only non-negative integral powers of x. So, it is a polynomial. Also, the highest power of x is 2, so, it is a polynomial of degree 2.

(vi) x2+2x1+3 is an expression having negative integral powers of x. So, it is not a polynomial.

(vii) 1. Clearly, 1 is a constant polynomial of degree 0.

(viii) 35 Clearly, 35 is a constant polynomial of degree 0.

(ix) x222x2 .  This is an expression having a negative integral power of x i.e. −2. So, it is not a polynomial.

(x) 22x28 is an expression having an only non-negative integral power of x. So, it is a polynomial. Also, the highest power of x is 2, so, it is a polynomial of degree 2.

(xi) 12x2 is an expression having a negative integral power of x. So, it is not a polynomial.

(xii) 15x12+1 .  In this expression, the power of x is 12 which is a fraction. Since it is an expression having a fractional power of x, so, it is not a polynomial.

(xiii) 35x273x+9 is an expression having only non-negative integral powers of x. So, it is a polynomial. Also, the highest power of x is 2, so, it is a polynomial of degree 2.

(xiv) x4x32+x3 In this expression, one of the powers of x is 32 which is a fraction. Since it is an expression having a fractional power of x, so, it is not a polynomial.

(xv) 2x3+3x2+x1 In this expression, one of the powers of x is 12 which is a fraction. Since it is an expression having a fractional power of x, so, it is not a polynomial.

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