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.