Question

Which of the following is a binomial in one variable of degree 5?
1. (x⁵ - 1)² - (x⁵ + 1)² + 5x⁵
2. (x5 + 1)2 + (x5 - 1)2
3. (x5 + 1)2 + (x5 - 1)2 -2(x¹⁰ - x<<br>4. (x5 - 1) (x5 + 1)

Answer 1

Correct Answer - Option 3 : (x5 + 1)2 + (x5 - 1)2 -2(x^{10
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.}