## Maharashtra State Board Class 9 Maths Solutions Chapter 3 Polynomials Practice Set 3.1

Question 1.

State whether the given algebraic expressions are polynomials? Justify.

i. y + \(\frac { 1 }{ y }\)

ii. 2 – 5√x

iii. x^{2} + 7x + 9

iv. 2m^{-2} + 7m – 5

v. 10

Answer:

i. No, because power of v in the term 5√x is -1 (negative number).

ii. No, because the power of x in the term 5√x is

i. e. 0.5 (decimal number).

iii. Yes. All the coefficients are real numbers. Also, the power of each term is a whole number.

iv. No, because the power of m in the term 2m^{-2} is -2 (negative number).

v. Yes, because 10 is a constant polynomial.

Question 2.

Write the coefficient of m^{3} in each of the given polynomial.

i. m^{3}

ii. \(\sqrt [ -3 ]{ 2 }\) + m – √3m^{3}

iii. \(\sqrt [ -2 ]{ 3 }\)m^{3} + 5m^{2} – 7m -1

Answer:

i. 1

ii. -√3

iii. – \(\frac { 2 }{ 3 }\)

Question 3.

Write the polynomial in x using the given information. [1 Mark each]

i. Monomial with degree 7

ii. Binomial with degree 35

iii. Trinomial with degree 8

Answer:

i. 5x^{7}

ii. x^{35} – 1

iii. 3x^{8} + 2x^{6} + x^{5}

Question 4.

Write the degree of the given polynomials.

i. √5

ii. x°

iii. x^{2}

iv. √2m^{10} – 7

v. 2p – √7

vi. 7y – y^{3} + y^{5}

vii. xyz +xy-z

viii. m^{3}n^{7} – 3m^{5}n + mn

Answer:

i. √5 = √5 x°

∴ Degree of the polynomial = 0

ii. x°

∴Degree of the polynomial = 0

iii. x^{2}

∴Degree of the polynomial = 2

iv. √2m^{10} – 7

Here, the highest power of m is 10.

∴Degree of the polynomial = 10

v. 2p – √7

Here, the highest power of p is 1.

∴ Degree of the polynomial = 1

vi. 7y – y^{3} + y^{5}

Here, the highest power of y is 5.

∴Degree of the polynomial = 5

vii. xyz + xy – z

Here, the sum of the powers of x, y and z in the term xyz is 1 + 1 + 1= 3,

which is the highest sum of powers in the given polynomial.

∴Degree of the polynomial = 3

viii. m^{3}n^{7} – 3m^{5}n + mn

Here, the sum of the powers of m and n in the term m^{3}n^{7} is 3 + 7 = 10,

which is the highest sum of powers in the given polynomial.

∴ Degree of the polynomial = 10

Question 5.

Classify the following polynomials as linear, quadratic and cubic polynomial. [2 Marks]

i. 2x^{2} + 3x +1

ii. 5p

iii. √2 – \(\frac { 1 }{ 2 }\)

iv. m^{3} + 7m^{2} + \(\sqrt [ 5 ]{ 2 }\)m – √7

v. a^{2}

vi. 3r^{3}

Answer:

Linear polynomials: ii, iii

Quadratic polynomials: i, v

Cubic polynomials: iv, vi

Question 6.

Write the following polynomials in standard form.

i. m^{3} + 3 + 5m

ii. – 7y + y^{5} + 3y^{3} – \(\frac { 1 }{ 2 }\)+ 2y^{4} – y^{2}

Answer:

i. m^{3} + 5m + 3

ii. y^{5} + 2y^{4} + 3y^{3} – y^{2} – 7y – \(\frac { 1 }{ 2 }\)

Question 7.

Write the following polynomials in coefficient form.

i. x^{3} – 2

ii. 5y

iii. 2m^{4} – 3m^{2} + 7

iv. – \(\frac { 2 }{ 3 }\)

Answer:

i. x^{3} – 2 = x^{3} + 0x^{2} + 0x – 2

∴ Coefficient form of the given polynomial = (1, 0, 0, -2)

ii. 5y = 5y + 0

∴Coefficient form of the given polynomial = (5,0)

iii. 2m^{4} – 3m^{2} + 7

= 2m^{4} + Om^{3} – 3m^{2} + 0m + 7

∴ Coefficient form of the given polynomial = (2, 0, -3, 0, 7)

iv. – \(\frac { 2 }{ 3 }\)

∴Coefficient form of the given polynomial = (- \(\frac { 2 }{ 3 }\))

Question 8.

Write the polynomials in index form.

i. (1, 2, 3)

ii. (5, 0, 0, 0 ,-1)

iii. (-2, 2, -2, 2)

Answer:

i. Number of coefficients = 3

∴ Degree = 3 – 1 = 2

∴ Taking x as variable, the index form is x^{2} + 2x + 3

ii. Number of coefficients = 5

∴ Degree = 5 – 1=4

∴ Taking x as variable, the index form is 5x^{4} + 0x^{3} + 0x^{2} + 0x – 1

iii. Number of coefficients = 4

∴Degree = 4 – 1 = 3

∴Taking x as variable, the index form is -2x^{3} + 2x^{2} – 2x + 2

Question 9.

Write the appropriate polynomials in the boxes.

Answer:

i. Quadratic polynomial: x^{2}; 2x^{2} + 5x + 10; 3x^{2} + 5x

ii. Cubic polynomial: x^{3} + x^{2} + x + 5; x^{3} + 9

iii. Linear polynomial: x + 7

iv. Binomial: x + 7; x^{3} + 9; 3x^{2} + 5x

v. Trinomial: 2x^{2} + 5x + 10

vi. Monomial: x^{2}

Question 1.

Write an example of a monomial, a binomial and a trinomial having variable x and degree 5. ( Textbook pg. no. 3)

Answer:

Monomial: x^{5}

Binomial: x^{5} + x

Trinomial: 2x^{5} – x^{2} + 5

Question 2.

Give example of a binomial in two variables having degree 5. (Textbook pg. no. 38)

Answer:

x^{3}y^{2} + xy