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 − x3/2 + 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 − x3/2 + x − 1 it is not a polynomial since the exponent of - x3/2 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