JEE Main 2022 (Online) 26th July Morning ShiftQuestion 1
The number of distinct real roots of the equation $x^5(x^3-x^2-x+1)+x(3x^3-4x^2-2x+4)-1=0$ is ________.
Quadratic Equation · Higher Degree Polynomial Equations
Answer$3$
Solution
Concept. Expand and factorise; the quintic factor is strictly increasing, so it has one real root. Formulas used. (1) If $g'(x)>0$ for all $x$, then $g$ has at most one real rootStep 1 - Factorise. $$x^8-x^7-x^6+x^5+3x^4-4x^3-2x^2+4x-1=(x-1)^2(x+1)(x^5+3x-1).$$ Step 2 - The quintic. $g(x)=x^5+3x-1$ has $g'=5x^4+3>0$. $g(0)=-1<0<g(1)=3$, so by (1) it has exactly one real root, in $(0,1)$; it is not $\pm1$. Step 3 - Count. Distinct real roots: $1$, $-1$ and the root of $g$. Three. Answer: 3