The number of distinct real roots of the equation \(x^7 – 7x – 2 = 0\) is
- 5
- 7
- 1
- 3
The Correct Option is D
Solution and Explanation
Given equation \(x^7 – 7x – 2 = 0\)
Let \(f(x) = x^7 – 7x – 2\)
\(f′(x) = 7x^6 – 7 = 7(x^6 – 1)\)
and \(f′(x) = 0 ⇒ x = +1\)
and \(f(–1) = –1 + 7 – 2 = 5 > 0\)
\(f(1) = 1 – 7 – 2 = –8 < 0\)
So, number of real roots of \(f(x) = 0\) and \(3\).
Hence, the correct option is (D): \(3\)
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
