CLASSIFY EACH POLYNOMIAL BY DEGREE AND NUMBER OF TERMS

Classifying polynomials based on number of terms.

Classifying polynomials based on degree.

Give the name for the following polynomials based on power and degree of the polynomials.

Problem 1 :

3

Solution :

Degree of the polynomial = 0, number of terms = 1

Name of the polynomial :

Based on degree = Constant 

Based on number of terms = Monomial

Problem 2 :

-p² + 2

Solution :

Degree of the polynomial = 2, number of terms = 2

Name of the polynomial :

Based on degree = Quadratic 

Based on number of terms = Binomial

Problem 3 :

6x⁴ - 2x³

Solution :

Degree of the polynomial = 4, number of terms = 2

Name of the polynomial :

Based on degree = Quartic 

Based on number of terms = Binomial

Problem 4 :

 5n⁴ - n² - 2

Solution :

Degree of the polynomial = 4, number of terms = 3

Name of the polynomial :

Based on degree = Quartic 

Based on number of terms = trinomial

Problem 5 :

-2p⁴ + 10p⁶ + 5p²

Solution :

Degree of the polynomial = 6, number of terms = 3

Name of the polynomial :

Based on degree = polynomial

Based on number of terms = trinomial

Problem 6 :

9m + 5m² + 10m³ - 5

Solution :

Degree of the polynomial = 3, number of terms = 3

Name of the polynomial :

Based on degree = cubic

Based on number of terms = polynomial

Problem 7 :

8n² - n³ + 7n

Solution :

Degree of the polynomial = 3, number of terms = 3

Name of the polynomial :

Based on degree = cubic

Based on number of terms = trinomial

Problem 8 :

For the polynomial P(x) = 5x³ - 3x² + 2x + √2, mark the statements as true or false and justify.

a) The degree of the polynomial P(x) is 4.

b) The degree of the polynomial P(x) is 3.

c) The coefficient x² is 3

d) The coefficient x² is 2

e) The constant is 3

f) The number of terms is 4.

Solution :

a) The degree is the highest exponent of the polynomial, for the given polynomial p(x) the highest exponent is 3. So, the given statement is false.

b) The degree of the polynomial P(x) is 3 and it is true.

c) The coefficient x² is -3, so the given statement is false.

d) It is false.

e) The constant is √2, so it is false.

f) The number of terms is 4, so it is true.

Problem 9 :

Justify the following statements with examples.

a) We can have a trinomial having degree 7.

b) The degree of a binomial cannot be more than two.

c) There is only one term of degree one in the monomial.

d) A cubic polynomial always has degree three

Solution :

a) 2x⁷ + 3x - 5

The highest exponent of the polynomial is 7.

Number of terms is 3.

b) 2x⁷ + 3x

It is binomial, the degree is 7. So, the given statement that the degree of a binomial cannot be more than two.

c) 3x and 5x²

These are some examples of monomials which has the degree 1 and 2 respectively. So, the given statement that "there is only one term of degree one in the monomial" is false.

d) Yes, a cubic polynomial always has degree three. 

Problem 10 :

Complete the entries 

p(x) = 5x⁷ - 6x⁵ + 7x - 6

Coefficient of x⁵ = 

Degree of p(x) = 

Constant term = 

Number of terms = 

Solution :

Coefficient of x⁵ = -6

Degree of p(x) = 7

Constant term = -6

Number of terms = 4

Problem 11 :

Which of the following is not a polynomial ?

a) x² + √2x + 3        b) x² - √2x + 6        c) x³ + 3x² - 3       d) 6x + 4

Solution :

To be polynomial, the highest exponent should not be a rational number and negative exponent.

In the above options, all of these are polynomial.

Problem 12 :

The degree of the polynomial 3x³ - x⁴ + 5x + 3 is 

a) 3    b)  -4     c) 4     d) 1

Solution :

By writing the given polynomial in standard form, we get

= - x⁴ + 3x³ + 5x + 3

The highest exponent is 4, so the degree of the polynomial is 4

Problem 13 :

Which of the following is a term of the polynomial ?

a) 2x     b) 3/x   c) √x    d) √x ^√x

Solution :

• 2x is the term of the polynomial
• 3/x is not the term of the polynomial, because the exponent is -1.
• √x is not the terms of the polynomial, because the exponent is 1/2 (rational number)
• √x ^√x is not the term of the polynomial.

Problem 14 :

If p(x) = 5 x² - 3x + 7, then p(1) equals

a)  -10    b) 9     c) -9    d)  10

Solution :

p(x) = 5 x² - 3x + 7

p(1) = 5 (1)² - 3(1) + 7

= 5 - 3 + 7

= 12 - 3

= 9

So, option b is correct.