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Which of the following is a binomial in one variable of degree 5?

Which of the following is a binomial in one variable of degree 5? Correct Answer (x<span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">5</span> + 1)<span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">2</span> + (x<span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">5</span> - 1)<span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">2</span> -2(x<sup>10</sup> - x<sup>5</sup>)

Concept:

Binomial: It is a type of polynomial that contains only two terms. It is the simplest kind of polynomial after the monomials (contains only one term).

Degree of a polynomial: It is the highest power of the variable of its individual terms.

Formula used: 

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

a2 - b2 = (a + b)(a - b)

Calculation:

Let's check from the option, one by one.

(1) (x5 - 1)2 - (x5 + 1)2 + 5x5

⇒ x10 - 2x5.1 + 1 - (x10 + 2x5.1 + 1) - 5x5

=  x10 - 2x5 + 1 - x10 - 2x5 - 1 - 5x5

= 5x5 

Which is monomial with degree 5.

(2) (x5 + 1)2 + (x5 - 1)2

= x10 + 2x5.1 + 1 + x10 - 2x5.1 + 1

= 2x10 + 2

Which is binomial with degree 10.

(3) (x5 + 1)2 + (x5 - 1)2 -2(x10 - x5)

= x10 + 2x5.1 + 1 + x10 - 2x5.1 + 1 - 2x10 + 2x5

= 2x5 + 2 

Which is binomial with degree 5.

(4) (x5 - 1) (x5 + 1)

= x10 - 1

Which is binomial with degree 10.

Hence, from the above discussion, we can see that only option 3 is correct.

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