**Question 1: Write the degree of each of the following polynomials:**

**(i) 2x ^{3} + 5x^{2} – 7**

**(ii) 5x ^{2} – 3x + 2**

**(iii) 2x + x ^{2} – 8**

**(iv) 1/2y ^{7} – 12y^{6} + 48y^{5} – 10**

**(v) 3x ^{3} + 1**

**(vi) 5**

**(vii) 20x ^{3} + 12x^{2}y^{2} – 10y^{2} + 20**

**Solution:**

In a polynomial, degree is the highest power of the variable.

(i)2x^{3}+ 5x^{2}– 7Given: 2x

^{3}+ 5x^{2}– 7Therefore, the degree of the polynomial, 2x

^{3}+ 5x^{2}– 7 is 3.

(ii)5x^{2}– 3x + 2Given: 5x

^{2}– 3x + 2Therefore, the degree of the polynomial, 5x

^{2}– 3x + 2 is 2.

(iii)2x + x^{2}– 8Given: 2x + x

^{2}– 8Therefore, the degree of the polynomial, 2x + x

^{2}– 8 is 2.

(iv)1/2y^{7}– 12y^{6}+ 48y^{5}– 10Given: 1/2y

^{7}– 12y^{6}+ 48y^{5}– 10Therefore, the degree of the polynomial, 1/2y

^{7}– 12y^{6}+ 48y^{5}– 10 is 7.

(v)3x^{3}+ 1Given: 3x

^{3}+ 1Therefore, the degree of the polynomial, 3x

^{3}+ 1 is 3

(vi)5Given: 5

Therefore, the degree of the polynomial, 5 is 0 as 5 is a constant number.

(vii)20x^{3}+ 12x^{2}y^{2}– 10y^{2}+ 20Given: 20x

^{3}+ 12x^{2}y^{2}– 10y^{2}+ 20Therefore, the degree of the polynomial, 20x

^{3}+ 12x^{2}y^{2}– 10y^{2}+ 20 is 4.

**Question 2: Which of the following expressions are not polynomials?**

**(i) x ^{2} + 2x^{-2}**

**(ii) √(ax) + x ^{2} – x^{3}**

**(iii) 3y ^{3} – √5y + 9**

**(iv) ax ^{1/2} + ax + 9x^{2} + 4**

**(v) 3x ^{-2} + 2x^{-1} + 4x + 5**

**Solution:**

(i)x^{2}+ 2x^{-2}Given: x

^{2}+ 2x^{-2}Since variable x has a power of -2 which is negative and as a polynomial does not contain any negative powers or fractions.

Therefore, the given expression is not a polynomial.

(ii) √(ax) + x^{2}– x^{3}Given: √(ax) + x

^{2}– x^{3}Since variable x has a power of 1/2 which is a fraction and as a polynomial does not contain any negative powers or fractions.

Therefore, the given expression is not a polynomial.

(iii) 3y^{3}– √5 y + 9Given: 3y

^{3}– √5 y + 9Since the polynomial has positive powers i.e. non-negative integers.

Therefore, the given expression is a polynomial.

(iv)ax^{1/2}+ ax + 9x^{2}+ 4Given: ax

^{1/2}+ ax + 9x^{2}+ 4Since variable x has a power of 1/2 which is a fraction and as a polynomial does not contain any negative powers or fractions.

Therefore, the given expression is not a polynomial.

(v)3x^{-2}+ 2x^{-1}+ 4x + 5Given: 3x

^{-2}+ 2x^{-1}+ 4x + 5Since variable x has a power of -2 and -1 which are negative and as a polynomial does not contain any negative powers or fractions.

The given expression is not a polynomial.

**Question 3: Write each of the following polynomials in the standard from. Also, write their degree:**

**(i) x ^{2} + 3 + 6x + 5x^{4}**

**(ii) a ^{2} + 4 + 5a^{6}**

**(iii) (x ^{3} – 1) (x^{3} – 4)**

**(iv) (y ^{3} – 2) (y^{3} + 11)**

**(v) (a ^{3} – 3/8) (a^{3} + 16/17)**

**(vi) (a + 3/4) (a + 4/3)**

**Solution:**

(i) x^{2}+ 3 + 6x + 5x^{4}Given: x

^{2}+ 3 + 6x + 5x^{4}Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.

Therefore, the expressions are:

(3 + 6x + x

^{2}+ 5x^{4}) or (5x^{4}+ x^{2}+ 6x + 3)The degree of the given polynomial is 4.

(ii) a^{2}+ 4 + 5a^{6}Given: a

^{2}+ 4 + 5a^{6}Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.

Therefore, the expressions are:

(4 + a

^{2}+ 5a^{6}) or (5a^{6}+ a^{2}+ 4)The degree of the given polynomial is 6.

(iii)(x^{3}– 1) (x^{3}– 4)Given: (x

^{3}– 1) (x^{3}– 4)x

^{6 }– 4x^{3}– x^{3}+ 4x

^{6}– 5x^{3}+ 4Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.

Therefore, the expressions are:

(4 – 5x

^{3}+ x^{6}) or (x^{6}– 5x^{3}+ 4)The degree of the given polynomial is 6.

(iv)(y^{3}– 2) (y^{3}+ 11)Given: (y

^{3}– 2) (y^{3}+ 11)y

^{6}+ 11y^{3}– 2y^{3}– 22y

^{6}+ 9y^{3}– 22Therefore, the expressions are:

(-22 + 9y

^{3}+ y^{6}) or (y^{6}+ 9y^{3}– 22)The degree of the given polynomial is 6.

(v)(a^{3}– 3/8) (a^{3}+ 16/17)Given: (a

^{3}– 3/8) (a^{3}+ 16/17)a

^{6}+ 16a^{3}/17 – 3a^{3}/8 – 6/17a

^{6}+ (77/136)a^{3}– 48/136Therefore, the expressions are:

(-48/136 + (77/136)a

^{3}+ a^{6}) or (a^{6}+ (77/136)a^{3}– 48/136)The degree of the given polynomial is 6.

(vi)(a + 3/4) (a + 4/3)Given: (a + 3/4) (a + 4/3)

a

^{2}+ 4a/3 + 3a/4 + 1a

^{2}+ (25/12)a + 1Therefore, the expressions are:

(1 + (25/12)a + a

^{2}) or (a^{2}+ (25/12)a + 1)The degree of the given polynomial is 2.

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