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

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

Factor the left side of the equation.

Factor $x$ out of $x^{4} - 5 x^{3} + x^{2} + 3 x$ .

Factor $x$ out of $x^{4}$ .

$x \cdot x^{3} - 5 x^{3} + x^{2} + 3 x = 0$

Factor $x$ out of $- 5 x^{3}$ .

$x \cdot x^{3} + x (- 5 x^{2}) + x^{2} + 3 x = 0$

Factor $x$ out of $x^{2}$ .

$x \cdot x^{3} + x (- 5 x^{2}) + x \cdot x + 3 x = 0$

Factor $x$ out of $3 x$ .

$x \cdot x^{3} + x (- 5 x^{2}) + x \cdot x + x \cdot 3 = 0$

Factor $x$ out of $x \cdot x^{3} + x (- 5 x^{2})$ .

$x (x^{3} - 5 x^{2}) + x \cdot x + x \cdot 3 = 0$

Factor $x$ out of $x (x^{3} - 5 x^{2}) + x \cdot x$ .

$x (x^{3} - 5 x^{2} + x) + x \cdot 3 = 0$

Factor $x$ out of $x (x^{3} - 5 x^{2} + x) + x \cdot 3$ .

$x (x^{3} - 5 x^{2} + x + 3) = 0$

Factor.

Factor $x^{3} - 5 x^{2} + x + 3$ using the rational roots test.

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.

$p = \pm 1, \pm 3 q = \pm 1$

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

$\pm 1, \pm 3$

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

Substitute $1$ into the polynomial.

$1^{3} - 5 \cdot 1^{2} + 1 + 3$

Raise $1$ to the power of $3$ .

$1 - 5 \cdot 1^{2} + 1 + 3$

Raise $1$ to the power of $2$ .

$1 - 5 \cdot 1 + 1 + 3$

Multiply $- 5$ by $1$ .

$1 - 5 + 1 + 3$

Subtract $5$ from $1$ .

$- 4 + 1 + 3$

Add $- 4$ and $1$ .

$- 3 + 3$

Add $- 3$ and $3$ .

$0$

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 5 x^{2} + x + 3}{x - 1}$

Divide $x^{3} - 5 x^{2} + x + 3$ by $x - 1$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{3}$

$5 x^{2}$

$x$

$3$

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			+	$x^{3}$	-	$x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - x^{2}$

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$x$

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$x$
					-	$4 x^{2}$	+	$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 4 x$

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	-	$4 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	-	$4 x$
							-	$3 x$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$4 x$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	-	$4 x$
							-	$3 x$	+	$3$

Divide the highest order term in the dividend $- 3 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	-	$3$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	-	$4 x$
							-	$3 x$	+	$3$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	-	$3$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	-	$4 x$
							-	$3 x$	+	$3$
							-	$3 x$	+	$3$

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x + 3$

				$x^{2}$	-	$4 x$	-	$3$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	-	$4 x$
							-	$3 x$	+	$3$
							+	$3 x$	-	$3$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$4 x$	-	$3$
$x$	-	$1$		$x^{3}$	-	$5 x^{2}$	+	$x$	+	$3$
			-	$x^{3}$	+	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	-	$4 x$
							-	$3 x$	+	$3$
							+	$3 x$	-	$3$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 4 x - 3$

Write $x^{3} - 5 x^{2} + x + 3$ as a set of factors.

$x ((x - 1) (x^{2} - 4 x - 3)) = 0$

Remove unnecessary parentheses.

$x (x - 1) (x^{2} - 4 x - 3) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 1 = 0$

$x^{2} - 4 x - 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x^{2} - 4 x - 3$ equal to $0$ and solve for $x$ .

Set $x^{2} - 4 x - 3$ equal to $0$ .

$x^{2} - 4 x - 3 = 0$

Solve $x^{2} - 4 x - 3 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot - 3)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 3}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 3}}{2 \cdot 1}$

Multiply $- 4$ by $- 3$ .

$x = \frac{4 \pm \sqrt{16 + 12}}{2 \cdot 1}$

Add $16$ and $12$ .

$x = \frac{4 \pm \sqrt{28}}{2 \cdot 1}$

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$x = \frac{4 \pm \sqrt{4 (7)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{7}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{7}}{2}$ .

$x = 2 \pm \sqrt{7}$

The final answer is the combination of both solutions.

$x = 2 + \sqrt{7}, 2 - \sqrt{7}$

The final solution is all the values that make $x (x - 1) (x^{2} - 4 x - 3) = 0$ true.

$x = 0, 1, 2 + \sqrt{7}, 2 - \sqrt{7}$

The result can be shown in multiple forms.

Exact Form:

$x = 0, 1, 2 + \sqrt{7}, 2 - \sqrt{7}$

Decimal Form:

$x = 0, 1, 4.64575131 \dots, - 0.64575131 \dots$

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

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