One root of the equation $x \left( 4 x ^ { 8 } - 3 \right) = 9 x$ is 0 . How many other roots are there?
A) 0
B) 7
C) 8
D) 9
E) none of these

Question

Quizplus · Accepted Answer

We can rearrange the equation as $4x^8-3x=9x$. This simplifies to $4x^8=12x$, or $x^7=3$. Since the coefficient of the $x^8$ term is positive, we know that this is an increasing function, and therefore has at most one real root. Since we already know that $x=0$ is one root, there can be at most 7 other roots. To see explicitly that there are 8 roots, note that $f(x)=x^7-3$ has exactly one real root by the intermediate value theorem (since $f(-1)=-4$ and $f(1)= -2$), and the other 7 roots are the complex roots of unity (i.e. the roots of $x^8=1$ other than $x=1$). Therefore, the answer is $\boxed{	extbf{(C) }8}$.

One Root of the Equation x(4x8−3)=9xx \left( 4 x ^ { 8 } - 3 \right) = 9 xx(4x8−3)=9x

One Root of the Equation $x \left( 4 x ^ { 8 } - 3 \right) = 9 x$