IDENTIFY INTO MONOMIAL BINOMIAL AND TRINOMIAL

Classify the following polynomials based on number of terms.

Problem 1 :

4y – 7z

Solution :

4y – 7z

Number of terms = 2

So, 4y – 7z is a binomial.

Problem 2 :

y²

Solution :

y²

Number of terms = 1

So, y² is a monomial.

Problem 3 :

x + y - xy

Solution :

x + y - xy

Number of terms = 3

So, x + y - xy is a trinomial.

Problem 4 :

100

Solution :

100

Number of terms = 1

So, 100 is a monomial.

Problem 5 :

ab – a - b

Solution :

ab - a - b

Number of terms = 3

So, ab - a - b is a trinomial.

Problem 6 :

5 – 3t

Solution :

5 – 3t

Number of terms = 2

So, 5 – 3t is a binomial.

Problem 7 :

4p²q – 4pq²

Solution :

4p²q – 4pq²

Number of terms = 2

So, 4p²q – 4pq² is a binomial.

Problem 8 :

7mn

Solution :

7mn

Number of terms = 1

So, 7mn is a monomial.

Problem 9 :

z² – 3z + 8

Solution :

z² – 3z + 8

Number of terms = 3

So, z² – 3z + 8 is a trinomial.

Problem 10 :

a² + b²

Solution :

a² + b²

Number of terms = 2

So, a² + b² is a binomial.

Problem 11 :

z² + z

Solution :

z² + z

Number of terms = 2

So, z² + z is a binomial.

Problem 12 :

1 + x + x²

Solution :

1 + x + x²

Number of terms = 3

So, 1 + x + x² is a trinomial.

Simplify the following by combining the like terms and then write whether the expression is a monomial, a binomial or a trinomial.

Problem 1 :

3x²yz² – 3xy²z + x²yz² + 7xy²z

Solution :

Given, 3x²yz² – 3xy²z + x²yz² + 7xy²z

= (3x²yz² + x²yz²) + (7xy²z – 3xy²z)

= (3 + 1) x²yz² + (7 – 3) xy²z

= 4x²yz² + 4xy²z

So, the expression is binomial.

Problem 2 :

x⁴ + 3x³y + 3x²y² – 3x³y – 3xy³ + y⁴ – 3x²y²

Solution :

x⁴ + 3x³y + 3x²y² – 3x³y – 3xy³ + y⁴ – 3x²y²

= 3x³y – 3x³y + 3x²y² – 3x²y² – 3xy³ + x⁴ + y⁴

= – 3xy³ + x⁴ + y⁴

So, the expression is trinomial.

Problem 3 :

p³q²r + pq²r³ + 3p²qr² – 9p²qr²

Solution :

p³q²r + pq²r³ + 3p²qr² – 9p²qr²

= p³q²r + pq²r³ – 6p²qr²

So, the expression is trinomial.

Problem 4 :

2a + 2b + 2c – 2a – 2b – 2c – 2b + 2c + 2a

Solution :

2a + 2b + 2c – 2a – 2b – 2c – 2b + 2c + 2a

= (2a – 2a + 2a) + (2b – 2b – 2b) + (2c – 2c + 2c)

= 2a – 2b + 2c

So, the expression is trinomial.

Problem 5 :

50x³ – 21x + 107 + 41x³ – x + 1 – 93 + 71x – 31x³

Solution :

= 50x³ – 21x + 107 + 41x³ – x + 1 – 93 + 71x – 31x³

= (50x³ + 41x³ – 31x³) + (-21x – x + 71x) + (107 + 1 – 93)

= 60x³ + 49x + 15

So, the expression is trinomial.

Problem 6 :

Answer the questions or identify the specified parts of the polynomial:

5x³ - 2x² + 14x + 7x² + 3x - 11

a. How many terms does this polynomial have? _______

b. Write the term that has a degree of 3. ________

c. What is the coefficient of this term with degree 3? _______

d. Which term is a constant term? _______

e. List a pair of like terms. ________ and ________

f. List another pair of like terms. ________ and ________

g. Rewrite the polynomial in simplified form by combining the like terms:

h. Evaluate this polynomial for = 2. _________

Solution :

a) Before combining the like terms, we have 6 terms.

b) Only one term which has the degree of 3.

c) Coefficient of the term with degree 3 is 5.

d) One term is constant which is -11.

e) -2x² and 7x² are like terms.

f) 14x and 3x are like terms.

g) By combining the like terms, we get

= 5x³ - 2x² + 7x² + 14x + 3x - 11

= 5x³ + 5x² + 17x - 11

h) When x = 2

f(2) = 5(2)³ + 5(2)² + 17(2) - 11

= 5(8) + 5(4) + 34 - 11

= 40 + 20 + 34 - 11

= 94 - 11

= 83

So, the answer is 83.

Problem 7 :

Answer the questions or identify the specified parts of the polynomial: 

2ab² + 3a² b - 6b - 2a² b + 3a - b

a. How many terms does this polynomial have? _______

b. Write the term(s) that have a degree of 1. ________________________

c. What are the coefficients of the term(s) with degree 1? _____________________

d. Which term is a constant term? _______

e. List a pair of like terms. ________ and ________

f. List another pair of like terms. ________ and ________

g. Rewrite the polynomial in simplified form by combining the like terms:

h. Evaluate this simplified polynomial for a = 4 and b = −1

Solution :

2ab² + 3a² b - 6b - 2a² b + 3a - b

a. Number of terms of the polynomial is 6.

b. Three terms are there which has the degree of 1.

c. Coefficient of 6b is 6

Coefficient of 3a is 3

Coefficient of -b is -1.

d. There is no constant.

e. 3a² b and  - 2a² b are like terms.

f. -6b and -b are like terms.

g. 2ab² + 3a² b - 6b - 2a² b + 3a - b

After combining the like terms, we get

= 2ab² + a² b - 7b + 3a

h. When a = 4 and b = −1

= 2(4)(-1)² + 4² (-1) - 7(-1) + 3(4)

= 8 - 16 + 7 + 12

= 27 - 16

= 11

So, the answer is 11.