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Physics Guru is an online education and competitive examinations preparation portal founded by Manish Verma, an IIT Madras alumnus, providing learning resources for competitive examinations.

The emphasis is on conceptual understanding, analytical thinking and problem solving rather than simply memorising formulas and standard methods.

The objective is to help students understand what they are learning and develop the ability to solve problems they have not seen before.

Solve for $x \in \mathbb{R}$

$7x^5 + x + 8 = 0$

-1 satisfies the given equation. Thus,

$7x^4 (x+1) - 7x^3 (x+1) + 7x^2 (x+1) -7x(x+1)+8(x+1)=0$

$\Rightarrow (x+1) (7x^4 - 7x^3 +7x^2 -7x +8) = 0$

Consider, $7x^4 - 7x^3 +7x^2 -7x +8 = 0$

Or, $7(x^4 - x^3 +x^2 -x) +8 = 0$

Let us consider III cases,

I. x < 0

Since $x^3$ and x both terms are negative, the left side is always +ve and cannot be 0. So, no solution is possible in this case.

II. x > 1

$x^4 > x^3$ and $x^2 > x$. So, left side is always +ve and cannot be 0. So, no solution is possible in this case.

III 0 < x < 1

$x^4 - x^3$ is a -ve decimal and so is $x^2 - x$. So, left side will be slightly smaller than 8 but cannot be 0. No solution in this case also.

As can be seen x cannot be 0. x cannot be 1 either.

The only solution possible is -1 for the original equation.