Solve for x x^3+2x-1=2

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

Step 1

Subtract $2$ from both sides of the equation.

$x^{3} + 2 x - 1 - 2 = 0$

Step 2

Subtract $2$ from $- 1$ .

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

Step 3

Factor $x^{3} + 2 x - 3$ 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.

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

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

$1 + 2 \cdot 1 - 3$

Multiply $2$ by $1$ .

$1 + 2 - 3$

Add $1$ and $2$ .

$3 - 3$

Subtract $3$ from $3$ .

$0$

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.

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

Divide $x^{3} + 2 x - 3$ 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 $0$ .

$x$

$1$

$x^{3}$

$0 x^{2}$

$2 x$

$3$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

$x^{2} + x + 3$

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

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

Step 4

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

$x - 1 = 0$

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

Step 5

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.

$x = 1$

Step 6

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

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

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

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

One to any power is one.

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

Multiply $- 4 \cdot 1 \cdot 3$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 12}}{2 \cdot 1}$

Subtract $12$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 11}}{2 \cdot 1}$

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{11}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{11}}{2}, - \frac{1 + i \sqrt{11}}{2}$

Step 7

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

$x = 1, - \frac{1 - i \sqrt{11}}{2}, - \frac{1 + i \sqrt{11}}{2}$

Solve for x x^3+2x-1=2

Related Questions

Support LCMGCF.com by disabling your adblocker.