UNDERSTANDING POLYNOMIALS

State whether the statements given are true or false.

Problem 1 :

1 + x/2 + x³ is a  polynomial.

Solution :

Given, the expression is 1 + x/2 + x³

Number of terms = 3

Therefore 1 + x/2 + x³ is a polynomial.

So, the statement is true.

Problem 2 :

(3a - b + 3) - (a + b) is a binomial.

Solution :

Given, the expression is (3a - b + 3) - (a + b)

= 3a - b + 3 - a - b

= 2a - 2b + 3

It is trinomial as there are 3 terms. So, the statement is false.

Problem 3 :

A trinomial can be a polynomial.

Solution :

In general, an expression with one or more than one term with the variables having integral powers is a polynomial.

In trinomials there are three terms with the variables having non negative integral powers, so it is a polynomial.

Hence, the given statement is true.

Problem 4 :

A polynomial with more than two terms is a trinomial.

Solution :

A polynomial with three unlike terms is called a trinomial.

Therefore, a polynomial with more than two terms needs not to be a trinomial.

Hence, the given statement is false.

Problem 5 :

Sum of x and y is x + y.

Solution :

Sum of x and y is x + y

So, the given statement is true.

Problem 6 :

Sum of 2 and p is 2p.

Solution :

Sum of 2 and p will be 2 + p.

2p is the product of 2 and p

So, the given statement is false.

Problem 7 :

A binomial has more than two terms.

Solution :

A polynomial with two unlike terms is a binomial.

Hence, the given statement is false.

Problem 8 :

A trinomial has exactly three terms.

Solution :

A polynomial with three unlike terms is a trinomial.

Therefore, a trinomial has exactly three terms.

Hence, the given statement is true.

Problem 9 :

In like terms, variables and their powers are the same.

Solution :

Like terms have same algebraic factors

Examples : 23x², 2x²

Therefore, in like terms, variables and their powers are the same.

Hence, the given statement is true.

Problem 10 :

The expression x + y + 5x is a trinomial.

Solution :

Given expression has two like terms x and 5x.

By adding like terms we get x + 5x = 6x

So, x + y + 5x = 6x + y which has two unlike terms.

Therefore, the expression is a binomial. So, the given statement is false.

Problem 11 :

4p is the numerical coefficient of q² in -4pq².

Solution :

The numerical factors of a term are called the numerical coefficient of the term.

-4 is the numerical coefficient of q² in -4pq²

So, 4p is not the numerical coefficient of q² in -4pq².

Hence, the given statement is false.

Problem 12 :

5a and 5b are unlike terms.

Solution :

5a and 5b have different algebraic factors.

Therefore, 5a and 5b are unlike terms.

Hence, the given statement is true.

Problem 13 :

If we add a monomial and binomial, then answer can never be a monomial.

Solution :

Consider a binomial -2xy + z and a monomial -z.

If we add -2xy + z and -z, we get

-2xy + z - z = -2xy, which is a monomial.

Therefore, if we add a monomial and binomial, the result can be a monomial.

Hence, the given statement is false.

Problem 14 :

If we subtract a monomial from a binomial, then answer is atleast a binomial.

Solution :

Consider a binomial -2xy + z and a monomial -z.

If we add -2xy + z and -z, we get

-2xy + z - z = -2xy, which is a monomial.

So, the result need not be a binomial.

Hence, the given statement is false.

Problem 15 :

When we subtract a monomial from a trinomial, then answer can be a polynomial.

Solution :

Consider a monomial x² and a trinomial x³ + 4x² + 5x.

Subtracting monomial from trinomial

= x³ + 4x² + 5x - x²

= x³ + 3x² + 5x, which is a polynomial.

Hence, the given statement is true.

Problem 16 :

The amount of money you have after investing $400 for 8 years and $600 for 6 years at the same interest rate is represented by 400x⁸ + 600x⁶, where x is the growth factor. Classify the polynomial by the number of terms. What is its degree?

Solution :

= 400x⁸ + 600x⁶

Number of terms of the polynomial is 2. So, it must be binomial.

Problem 17 :

The expression (4/3) πr³ represents the volume of a sphere with radius r. Why is this expression a monomial? What is its degree?

Solution :

= (4/3) πr³

The terms can be separated by positive or negative sign, since we don't have any sign in between them, it can be considered as one term.

So, it is monomial.

Problem 18 :

The number of individual memberships at a fitness center in m months is represented by 142 + 12m. The number of family memberships at the fitness center in m months is represented by 52 + 6m. Write a polynomial that represents the total number of memberships at the fitness center.

Solution :

Cost for fitness center for m months for individual membership,

= 142 + 12m

Cost for fitness center for m months for family membership,

= 52 + 6m

Total cost = 142 + 12m + 52 + 6m

= 142 + 52 + 12m + 6m

= 194 + 18m