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#### EXERCISE 8.1 PAGE NO: 8.2

**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:**

**(i)** 2x^{3} + 5x^{2} – 7

We know that in a polynomial, degree is the highest power of the variable.

The degree of the polynomial, 2x^{3} + 5x^{2} – 7 is 3.

**(ii)** 5x^{2} – 3x + 2

The degree of the polynomial, 5x^{2} – 3x + 2 is 2.

**(iii)** 2x + x^{2} – 8

The degree of the polynomial, 2x + x^{2} – 8 is 2.

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

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

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

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

**(vi)** 5

The degree of the polynomial, 5 is 0 (since 5 is a constant number).

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

The degree of the polynomial, 20x^{3} + 12x^{2}y^{2} – 10y^{2} + 20 is 4.

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

**Solution:**

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

The given expression is not a polynomial.

Because a polynomial does not contain any negative powers or fractions.

**(ii)** √(ax) + x^{2} – x^{3}

The given expression is a polynomial.

Because the polynomial has positive powers.

**(iii)** 3y^{3} – √5y + 9

The given expression is a polynomial.

Because the polynomial has positive powers.

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

The given expression is not a polynomial.

Because a polynomial does not contain any negative powers or fractions.

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

The given expression is not a polynomial.

Because a polynomial does not contain any negative powers or fractions.

**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}

The standard form of the polynomial is written in either increasing or decreasing order of their powers.

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}

The standard form of the polynomial is written in either increasing or decreasing order of their powers.

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)

x^{6} – 4x^{3} – x^{3} + 4

x^{6} – 5x^{3} + 4

The standard form of the polynomial is written in either increasing or decreasing order of their powers.

x^{6} – 5x^{3} + 4 or 4 – 5x^{3} + x^{6}

The degree of the given polynomial is 6.

**(iv)** (y^{3} – 2) (y^{3} + 11)

y^{6} + 11y^{3} – 2y^{3} – 22

y^{6} + 9y^{3} – 22

y^{6} + 9y^{3} – 22 or -22 + 9y^{3} + y^{6}

The degree of the given polynomial is 6.

**(v)** (a^{3} – 3/8) (a^{3} + 16/17)

a^{6} + 16a^{3}/17 – 3a^{3}/8 – 6/17

a^{6} + 77/136a^{3} – 48/136

a^{6} + 77/136a^{3} – 48/136 or -48/136 + 77/136a^{3} + a^{6}

The degree of the given polynomial is 6.

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

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

a^{2} + 25a/12 + 1

a^{2} + 25a/12 + 1 or 1 + 25a/12 + a^{2}

The degree of the given polynomial is 2.

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