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  • Write whether the following statements are True or False. Justify your answer. i. A binomial can have at most two terms ii. Every polynomial is a binomial. iii. A binomial may have degree 5. iv. Zero of a polynomial is always 0

Write whether the following statements are True or False. Justify your answer.

i. A binomial can have at most two terms

ii. Every polynomial is a binomial.

iii. A binomial may have degree 5.

iv. Zero of a polynomial is always 0

v.A polynomial cannot have more than one zero.

vi. The degree of the sum of two polynomials each of degree 5 is always 5

Answers (1)

i.  False

Solution :-  Binomial: A binomial is an expression that has two numbers, terms or letters joined by the sign + or –.

Binomial necessarily means consisting of two terms only. These terms should not be like terms.

For example: x^{2}+1  is a binomial

x + 2x is not a binomial as these are like terms.

So the given statement is false, as a binomial has exactly two terms (not at most two terms).

ii. False

Solution:- Binomial: A binomial is an expression that has two numbers, terms or letters joined by the sign  + or –.

Binomial necessarily means consisting of two terms only. These terms should not be like terms.

For example: x^{2}+1  is a binomial

x + 2x is not a binomial as these are like terms.

Polynomial: It is an expression of more than two algebraic terms, especially the sum of several  terms that contains different powers of the same variable(s)

Its degree is always a whole number.

For example:  x^{0},x^{2}+2  etc.

Because a binomial has exactly two terms but a polynomial can be monomial (single term), binomial (two terms), trinomial (three terms) etc.

\therefore  The given statement is False.

iii. True

Solution:-  Binomial: A binomial is an expression that has two numbers, terms or letters joined by the sign  + or –.

Binomial necessarily means consisting of two terms only. These terms should not be like terms.

For example: x^{2}+1  is a binomial

x + 2x is not a binomial as these are like terms.

Degree of polynomial: Degree of polynomial is the highest power of the polynomial’s monomials with non-zero coefficient.

For any binomial of the form x^{5}+2, we can see that the degree is 5.

So, a binomial may have degree 5.

Therefore the given statement is True.

iv. False

Solution :- Polynomial:- It is an expression of more than two algebraic terms, especially the sum of several  terms that contains different powers of the same variable(s)

Its degree is always a whole number.

For example:  x^{0},x^{2}+2  etc.

We know that for finding the zero of a polynomial, we need to find a value of x for which the polynomial will be zero

i.e., p(x)=0

Let us consider an example:

p(x) = x - 2

Now to find the zero of this polynomial, we have:

x - 2 =0

x = 2                            (which is not zero)

Hence the given statement is false because zero of a polynomial can be any real number.

v.  False

Solution :-  Polynomial: It is an expression of more than two algebraic terms, especially the sum of several  terms that contains different powers of the same variable(s).

Its degree is always a whole number.

For example:  x^{0},x+2,x^{3}+1,x^{4}+x^{3}+x^{2}+1  etc.

We know that for finding the zero of a polynomial, we need to find a value of x for which the polynomial will be zero

i.e., p(x)=0

So, a polynomial can have any number of zeroes. It depends upon the degree of polynomial.

\therefore  The given statement is False.

vi. False

Solution :-  Degree of polynomial: Degree of polynomial is the highest power of the polynomial’s monomials with non-zero coefficient.

The degree of the sum of two polynomials may be less than or equal to 5.

For example: x^{5}+1  and -x^{5}+2x^{3}+1  are two polynomials of degree 5 but the degree of the sum of the two polynomials 2x^{3}+2 is 3.

Hence the given statement is False.

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