Find the Roots (Zeros) x^5+x^4+x^3+x^2+x=0

$x^{5} + x^{4} + x^{3} + x^{2} + x = 0$

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 0$

$q = \pm 1$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$0$

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(0)}^{5} + {(0)}^{4} + {(0)}^{3} + {(0)}^{2} + 0$

Simplify the expression. In this case, the expression is equal to $0$ so $x = 0$ is a root of the polynomial.

Remove parentheses.

${(0)}^{5} + {(0)}^{4} + {(0)}^{3} + {(0)}^{2} + 0$

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$0 + {(0)}^{4} + {(0)}^{3} + {(0)}^{2} + 0$

Raising $0$ to any positive power yields $0$ .

$0 + 0 + {(0)}^{3} + {(0)}^{2} + 0$

Raising $0$ to any positive power yields $0$ .

$0 + 0 + 0 + {(0)}^{2} + 0$

Raising $0$ to any positive power yields $0$ .

$0 + 0 + 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$0 + 0 + 0 + 0$

Add $0$ and $0$ .

$0 + 0 + 0$

Add $0$ and $0$ .

$0 + 0$

Add $0$ and $0$ .

$0$

Since $0$ is a known root, divide the polynomial by $x + 0$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{5} + x^{4} + x^{3} + x^{2} + x}{x + 0}$

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$0$	$1$	$1$	$1$	$1$	$1$	$0$
	\xa0					\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$0$	$1$	$1$	$1$	$1$	$1$	$0$
	\xa0					\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(1)$ .

$0$	$1$	$1$	$1$	$1$	$1$	$0$
	\xa0	$0$				\xa0
	$1$