Chapter 6: Factorization of Polynomials Exercise – 6.1

Question: 1

Which of the following expressions are polynomials in one variable and which are not?

State the reasons for your answers

1. 3x² − 4x + 15

2. y² + 2√3

3. 3√x + √2 x

4. x − 4/x

5. x¹² + y² + t⁵⁰

Solution:

1. 3x² − 4x + 15 it is a polynomial of x

2. y² + 2√3 it is a polynomial of y

 3√x + √2 x it is not a polynomial since the exponent of 3√x is not a positive term

4. x − 4/x- it is not a polynomial since the exponent of - 4/x is not a positive term

5. x¹² + y² + t⁵⁰ it is a three variable polynomial which variables of x, y, t

Question: 2

Write the coefficients of x² in each of the following

1. 17 − 2x + 7x²

2. 9 − 12x + x²

4. √3 x – 7

Solution:

Given, to find the coefficients of x²

1. 17 − 2x + 7x² - the coefficient is 7

2. 9 − 12x + x² - the coefficient is 0

- the coefficient is π/6 

4. √3 x - 7 - the coefficient is 0

Question: 3

Write the degrees of each of the following polynomials:

1. 7x³ + 4x² − 3x + 12

2. 12 − x + 2x²

3.  5y - √2

4. 7 − 7x^°

5. 0

Solution:

Given, to find degrees of the polynomials

Degree is highest power in the polynomial

1. 7x³ + 4x² − 3x + 12 the degree is 3

2. 12 − x + 2x² the degree is 3

3. 5y - √2 the degree is 1

4. 7 − 7x° the degree is 0

5. 0 the degree of 0 is not defined

Question: 4

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

1. x + x² + 4

2. 3x - 2

3. 2x + x²

4. 3y

5. t² + 1

f. 7t⁴ + 4t² + 3t − 2

Solution:

Given

1. x + x² + 4 - it is a quadratic polynomial as its degree is 2

2. 3x – 2 - it is a linear polynomial as its degree is 1

3. 2x + x² - it is a quadratic polynomial as its degree is 2

4. 3y - it is a linear polynomial as its degree is 1

5. t² + 1- it is a quadratic polynomial as its degree is 2

f. 7t⁴ + 4t² + 3t – 2 - it is a bi-quadratic polynomial as its degree is 4

Question: 5

Classify the following polynomials as polynomials in one variables, two - variables etc:

1. x² − xy + 7y²

2. x² − 2tx + 7t² − x + t

3. t³ − 3t² + 4t − 5

4. xy + yz + zx

Solution:

Given

1. x² − xy + 7y² it is a polynomial in two variables x and y

2. x² − 2tx + 7t² − x + t it is a polynomial in two variables x and t

3. t³ − 3t² + 4t − 5 it is a polynomial in one variable t

4. xy + yz + zx it is a polynomial in 3 variables in x , y and z

Question: 6

Identify the polynomials in the following:

1. f(x) = 4x³ − x² − 3x + 7

4. q(x) = 2x² − 3x + 4/x + 2

5. h(x) = x⁴ − x^3/2 + x − 1

6. f(x) = 2 + 3x + 4x

Solution:

Given

1. f(x) = 4x³ − x² − 3x + 7 it is a polynomial

it is not a polynomial since the exponent of √x is a negative integer

it is a polynomial as it has positive integers as exponents

4. q(x) = 2x² − 3x + 4/x + 2 it is not a polynomial since the exponent of 4/x is a negative integer

5. h(x) = x⁴ − x^3/2 + x − 1 it is not a polynomial since the exponent of - x^3/2 is a negative integer

6. f(x) = 2 + 3x + 4x it is not a polynomial since the exponent of 3/x is a negative integer

Question: 7

Identify constant, linear, quadratic abd cubic polynomial from the following polynomials :

1. f(x) = 0

2. g(x) = 2x³ − 7x + 4

3. h(x) = −3x + 1/2

4. p(x) = 2x² − x + 4

5. q(x) = 4x + 3

6. r(x) = 3x³ + 4x² + 5x − 7

Solution:

Given,

1. f(x) = 0 as 0 is constant , it is a constant variable

2. g(x) = 2x³ − 7x + 4 since the degree is 3, it is a cubic polynomial

3. h(x) = −3x + 1/2 since the degree is 1, it is a linear polynomial

4. p(x) = 2x² − x + 4 since the degree is 2, it is a quadratic polynomial

5. q(x) = 4x + 3 since the degree is 1, it is a linear polynomial

6. r(x) = 3x³ + 4x² + 5x − 7 since the degree is 3, it is a cubic polynomial

Question: 8

Give one example each of a binomial of degree 25, and of a monomial of degree 100

Solution:

Given, to write the examples for binomial and monomial with the given degrees

Example of a binomial with degree 25 - 7x³⁵ − 5

Example of a monomial with degree 100 - 2t¹⁰⁰