Question

# Which of the following expressions are polynomials? In case of a polynomial, write its degree.  (i) x5−2x3+x+√3 (ii) y3+√3y (iii) t2−25t+√5 (iv) x1000−1 (v) 1√2x2−√2x+2 (vi) x−2+2x−1+3 (vii) 1 (viii) −35 (ix) x22−2x2 (x) 2√2x2−8 (xi) 12x2 (xii) 1√5x12+1 (xiii) 35x2−73x+9 (xiv) x4−x32+x−3 (xv) 2x3+3x2+√x−1

Solution

## (i) x5−2x3+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) t2−25t+√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) x1000−1 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) 1√2x2−√2x+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) x−2+2x−1+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) x22−2x2 .  This is an expression having a negative integral power of x i.e. −2. So, it is not a polynomial. (x) 2√2x2−8 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) 1√5x12+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) 35x2−73x+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) x4−x32+x−3 . 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+√x−1 . 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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