Find the Roots (Zeros) 5x^3-4x+1=0

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

Step 1

Factor $5 x^{3} - 4 x + 1$ 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.

$p = \pm 1 q = \pm 1, \pm 5$

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

$\pm 1, \pm 0.2$

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.

$5 {(- 1)}^{3} - 4 \cdot - 1 + 1$

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

$5 \cdot - 1 - 4 \cdot - 1 + 1$

Multiply $5$ by $- 1$ .

$- 5 - 4 \cdot - 1 + 1$

Multiply $- 4$ by $- 1$ .

$- 5 + 4 + 1$

Add $- 5$ and $4$ .

$- 1 + 1$

Add $- 1$ and $1$ .

$0$

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.

$\frac{5 x^{3} - 4 x + 1}{x + 1}$

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

$x$

$1$

$5 x^{3}$

$0 x^{2}$

$4 x$

$1$

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x$ .

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

Multiply the new quotient term by the divisor.

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

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} + 5 x^{2}$

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

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

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

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

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

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

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

Multiply the new quotient term by the divisor.

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

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} - 5 x$

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

The expression needs to be subtracted from the dividend, so change all the signs in $x + 1$

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

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

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

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

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

Write $5 x^{3} - 4 x + 1$ as a set of factors.

$(x + 1) (5 x^{2} - 5 x + 1) = 0$

Step 2

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

$x + 1 = 0$

$5 x^{2} - 5 x + 1 = 0$

Step 3

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

Set $5 x^{2} - 5 x + 1$ equal to $0$ and solve for $x$ .

Set $5 x^{2} - 5 x + 1$ equal to $0$ .

$5 x^{2} - 5 x + 1 = 0$

Solve $5 x^{2} - 5 x + 1 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 5$ , $b = - 5$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (5 \cdot 1)}}{2 \cdot 5}$

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 1$ .

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20 \cdot 1}}{2 \cdot 5}$

Multiply $- 20$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 20}}{2 \cdot 5}$

Subtract $20$ from $25$ .

$x = \frac{5 \pm \sqrt{5}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{5 \pm \sqrt{5}}{10}$

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{5}}{10}, \frac{5 - \sqrt{5}}{10}$

Step 5

The final solution is all the values that make $(x + 1) (5 x^{2} - 5 x + 1) = 0$ true.

$x = - 1, \frac{5 + \sqrt{5}}{10}, \frac{5 - \sqrt{5}}{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - 1, \frac{5 + \sqrt{5}}{10}, \frac{5 - \sqrt{5}}{10}$

Decimal Form:

$x = - 1, 0.72360679 \dots, 0.27639320 \dots$

Step 7

Find the Roots (Zeros) 5x^3-4x+1=0

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