Solve for x x^3-4/x=6

$x^{3} - \frac{4}{x} = 6$

Step 1

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1$

The LCM of one and any expression is the expression.

$x$

Step 2

Multiply each term in $x^{3} - \frac{4}{x} = 6$ by $x$ to eliminate the fractions.

Multiply each term in $x^{3} - \frac{4}{x} = 6$ by $x$ .

$x^{3} x - \frac{4}{x} x = 6 x$

Simplify the left side.

Simplify each term.

Multiply $x^{3}$ by $x$ by adding the exponents.

Multiply $x^{3}$ by $x$ .

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

$x^{3} x^{1} - \frac{4}{x} x = 6 x$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3 + 1} - \frac{4}{x} x = 6 x$

Add $3$ and $1$ .

$x^{4} - \frac{4}{x} x = 6 x$

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{4}{x}$ into the numerator.

$x^{4} + \frac{- 4}{x} x = 6 x$

Cancel the common factor.

$x^{4} + \frac{- 4}{x} x = 6 x$

Rewrite the expression.

$x^{4} - 4 = 6 x$

Step 3

Solve the equation.

Reorder the polynomial.

$x^{4} - 6 x - 4$

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.

$p = \pm 1, \pm 2, \pm 4$

$q = \pm 1$

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

$\pm 1, \pm 2, \pm 4$

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.

${(2)}^{4} - 4 - 6 \cdot 2$

Simplify the expression. In this case, the expression is equal to $0$ so $x = 2$ is a root of the polynomial.

Simplify each term.

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

$16 - 4 - 6 \cdot 2$

Multiply $- 6$ by $2$ .

$16 - 4 - 12$

Simplify by subtracting numbers.

Subtract $4$ from $16$ .

$12 - 12$

Subtract $12$ from $12$ .

$0$

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 4 - 6 x}{x - 2}$

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.

$2$	$1$	$0$	$0$	$- 6$	$- 4$
	\xa0				\xa0

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

$2$	$1$	$0$	$0$	$- 6$	$- 4$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(2)$ and place the result of $(2)$ under the next term in the dividend $(0)$ .

$2$	$1$	$0$	$0$	$- 6$	$- 4$
	\xa0	$2$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$1$	$0$	$0$	$- 6$	$- 4$
	\xa0	$2$			\xa0
	$1$	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(2)$ and place the result of $(4)$ under the next term in the dividend $(0)$ .

$2$	$1$	$0$	$0$	$- 6$	$- 4$
	\xa0	$2$	$4$		\xa0
	$1$	$2$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$1$	$0$	$0$	$- 6$	$- 4$
	\xa0	$2$	$4$		\xa0
	$1$	$2$	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(2)$ and place the result of $(8)$ under the next term in the dividend $(- 6)$ .

$2$	$1$	$0$	$0$	$- 6$	$- 4$
	\xa0	$2$	$4$	$8$	\xa0
	$1$	$2$	$4$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$1$	$0$	$0$	$- 6$	$- 4$
	\xa0	$2$	$4$	$8$	\xa0
	$1$	$2$	$4$	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(2)$ and place the result of $(4)$ under the next term in the dividend $(- 4)$ .

$2$	$1$	$0$	$0$	$- 6$	$- 4$
	\xa0	$2$	$4$	$8$	$4$
	$1$	$2$	$4$	$2$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$1$	$0$	$0$	$- 6$	$- 4$
	\xa0	$2$	$4$	$8$	$4$
	$1$	$2$	$4$	$2$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 2 x^{2} + (4) x + 2$

Simplify the quotient polynomial.

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

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.63889691$

The polynomial can be written as a set of linear factors.

$(x - 2) (x + 0.63889691)$

These are the roots (zeros) of the polynomial $x^{4} - 6 x - 4$ .

$x = 2, - 0.63889691$

Solve for x x^3-4/x=6

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