Find All Complex Solutions 4x^4-18x^3+42x^2-108x+108=0

$4 x^{4} - 18 x^{3} + 42 x^{2} - 108 x + 108 = 0$

Step 1

Factor the left side of the equation.

Factor $2$ out of $4 x^{4} - 18 x^{3} + 42 x^{2} - 108 x + 108$ .

Factor $2$ out of $4 x^{4}$ .

$2 (2 x^{4}) - 18 x^{3} + 42 x^{2} - 108 x + 108 = 0$

Factor $2$ out of $- 18 x^{3}$ .

$2 (2 x^{4}) + 2 (- 9 x^{3}) + 42 x^{2} - 108 x + 108 = 0$

Factor $2$ out of $42 x^{2}$ .

$2 (2 x^{4}) + 2 (- 9 x^{3}) + 2 (21 x^{2}) - 108 x + 108 = 0$

Factor $2$ out of $- 108 x$ .

$2 (2 x^{4}) + 2 (- 9 x^{3}) + 2 (21 x^{2}) + 2 (- 54 x) + 108 = 0$

Factor $2$ out of $108$ .

$2 (2 x^{4}) + 2 (- 9 x^{3}) + 2 (21 x^{2}) + 2 (- 54 x) + 2 (54) = 0$

Factor $2$ out of $2 (2 x^{4}) + 2 (- 9 x^{3})$ .

$2 (2 x^{4} - 9 x^{3}) + 2 (21 x^{2}) + 2 (- 54 x) + 2 (54) = 0$

Factor $2$ out of $2 (2 x^{4} - 9 x^{3}) + 2 (21 x^{2})$ .

$2 (2 x^{4} - 9 x^{3} + 21 x^{2}) + 2 (- 54 x) + 2 (54) = 0$

Factor $2$ out of $2 (2 x^{4} - 9 x^{3} + 21 x^{2}) + 2 (- 54 x)$ .

$2 (2 x^{4} - 9 x^{3} + 21 x^{2} - 54 x) + 2 (54) = 0$

Factor $2$ out of $2 (2 x^{4} - 9 x^{3} + 21 x^{2} - 54 x) + 2 (54)$ .

$2 (2 x^{4} - 9 x^{3} + 21 x^{2} - 54 x + 54) = 0$

Regroup terms.

$2 (2 x^{4} - 54 x - 9 x^{3} + 21 x^{2} + 54) = 0$

Factor $2 x$ out of $2 x^{4} - 54 x$ .

Factor $2 x$ out of $2 x^{4}$ .

$2 (2 x (x^{3}) - 54 x - 9 x^{3} + 21 x^{2} + 54) = 0$

Factor $2 x$ out of $- 54 x$ .

$2 (2 x (x^{3}) + 2 x (- 27) - 9 x^{3} + 21 x^{2} + 54) = 0$

Factor $2 x$ out of $2 x (x^{3}) + 2 x (- 27)$ .

$2 (2 x (x^{3} - 27) - 9 x^{3} + 21 x^{2} + 54) = 0$

Rewrite $27$ as $3^{3}$ .

$2 (2 x (x^{3} - 3^{3}) - 9 x^{3} + 21 x^{2} + 54) = 0$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 3$ .

$2 (2 x ((x - 3) (x^{2} + x \cdot 3 + 3^{2})) - 9 x^{3} + 21 x^{2} + 54) = 0$

Factor.

Simplify.

Move $3$ to the left of $x$ .

$2 (2 x ((x - 3) (x^{2} + 3 x + 3^{2})) - 9 x^{3} + 21 x^{2} + 54) = 0$

Raise $3$ to the power of $2$ .

$2 (2 x ((x - 3) (x^{2} + 3 x + 9)) - 9 x^{3} + 21 x^{2} + 54) = 0$

Remove unnecessary parentheses.

$2 (2 x (x - 3) (x^{2} + 3 x + 9) - 9 x^{3} + 21 x^{2} + 54) = 0$

Factor $3$ out of $- 9 x^{3} + 21 x^{2} + 54$ .

Factor $3$ out of $- 9 x^{3}$ .

$2 (2 x (x - 3) (x^{2} + 3 x + 9) + 3 (- 3 x^{3}) + 21 x^{2} + 54) = 0$

Factor $3$ out of $21 x^{2}$ .

$2 (2 x (x - 3) (x^{2} + 3 x + 9) + 3 (- 3 x^{3}) + 3 (7 x^{2}) + 54) = 0$

Factor $3$ out of $54$ .

$2 (2 x (x - 3) (x^{2} + 3 x + 9) + 3 (- 3 x^{3}) + 3 (7 x^{2}) + 3 (18)) = 0$

Factor $3$ out of $3 (- 3 x^{3}) + 3 (7 x^{2})$ .

$2 (2 x (x - 3) (x^{2} + 3 x + 9) + 3 (- 3 x^{3} + 7 x^{2}) + 3 (18)) = 0$

Factor $3$ out of $3 (- 3 x^{3} + 7 x^{2}) + 3 (18)$ .

$2 (2 x (x - 3) (x^{2} + 3 x + 9) + 3 (- 3 x^{3} + 7 x^{2} + 18)) = 0$

Factor.

Factor $- 3 x^{3} + 7 x^{2} + 18$ 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.

$p = \pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6 q = \pm 1, \pm 3$

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

$\pm 1, \pm 0.\overline{3}, \pm 18, \pm 6, \pm 2, \pm 0.\overline{6}, \pm 9, \pm 3$

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

Substitute $3$ into the polynomial.

$- 3 \cdot 3^{3} + 7 \cdot 3^{2} + 18$

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

$- 3 \cdot 27 + 7 \cdot 3^{2} + 18$

Multiply $- 3$ by $27$ .

$- 81 + 7 \cdot 3^{2} + 18$

Raise $3$ to the power of $2$ .

$- 81 + 7 \cdot 9 + 18$

Multiply $7$ by $9$ .

$- 81 + 63 + 18$

Add $- 81$ and $63$ .

$- 18 + 18$

Add $- 18$ and $18$ .

$0$

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 3 x^{3} + 7 x^{2} + 18}{x - 3}$

Divide $- 3 x^{3} + 7 x^{2} + 18$ by $x - 3$ .

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

$x$

$3$

$3 x^{3}$

$7 x^{2}$

$0 x$

$18$

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

				-	$3 x^{2}$
	$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			-	$3 x^{3}$	+	$9 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{3} + 9 x^{2}$

			-	$3 x^{2}$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$

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

			-	$3 x^{2}$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$

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

			-	$3 x^{2}$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$	+	$0 x$

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

			-	$3 x^{2}$	-	$2 x$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$	+	$0 x$

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$	-	$2 x$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					-	$2 x^{2}$	+	$6 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} + 6 x$

			-	$3 x^{2}$	-	$2 x$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$6 x$

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

			-	$3 x^{2}$	-	$2 x$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$6 x$
							-	$6 x$

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

			-	$3 x^{2}$	-	$2 x$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$6 x$
							-	$6 x$	+	$18$

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

			-	$3 x^{2}$	-	$2 x$	-	$6$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$6 x$
							-	$6 x$	+	$18$

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$	-	$2 x$	-	$6$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$6 x$
							-	$6 x$	+	$18$
							-	$6 x$	+	$18$

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x + 18$

			-	$3 x^{2}$	-	$2 x$	-	$6$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$6 x$
							-	$6 x$	+	$18$
							+	$6 x$	-	$18$

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

			-	$3 x^{2}$	-	$2 x$	-	$6$
$x$	-	$3$	-	$3 x^{3}$	+	$7 x^{2}$	+	$0 x$	+	$18$
			+	$3 x^{3}$	-	$9 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$6 x$
							-	$6 x$	+	$18$
							+	$6 x$	-	$18$
										$0$

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

$- 3 x^{2} - 2 x - 6$

Write $- 3 x^{3} + 7 x^{2} + 18$ as a set of factors.

$2 (2 x (x - 3) (x^{2} + 3 x + 9) + 3 ((x - 3) (- 3 x^{2} - 2 x - 6))) = 0$

Remove unnecessary parentheses.

$2 (2 x (x - 3) (x^{2} + 3 x + 9) + 3 (x - 3) (- 3 x^{2} - 2 x - 6)) = 0$

Factor $x - 3$ out of $2 x (x - 3) (x^{2} + 3 x + 9) + 3 (x - 3) (- 3 x^{2} - 2 x - 6)$ .

Factor $x - 3$ out of $2 x (x - 3) (x^{2} + 3 x + 9)$ .

$2 ((x - 3) (2 x (x^{2} + 3 x + 9)) + 3 (x - 3) (- 3 x^{2} - 2 x - 6)) = 0$

Factor $x - 3$ out of $3 (x - 3) (- 3 x^{2} - 2 x - 6)$ .

$2 ((x - 3) (2 x (x^{2} + 3 x + 9)) + (x - 3) (3 (- 3 x^{2} - 2 x - 6))) = 0$

Factor $x - 3$ out of $(x - 3) (2 x (x^{2} + 3 x + 9)) + (x - 3) (3 (- 3 x^{2} - 2 x - 6))$ .

$2 ((x - 3) (2 x (x^{2} + 3 x + 9) + 3 (- 3 x^{2} - 2 x - 6))) = 0$

Apply the distributive property.

$2 ((x - 3) (2 x \cdot x^{2} + 2 x (3 x) + 2 x \cdot 9 + 3 (- 3 x^{2} - 2 x - 6))) = 0$

Simplify.

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$2 ((x - 3) (2 (x^{2} x) + 2 x (3 x) + 2 x \cdot 9 + 3 (- 3 x^{2} - 2 x - 6))) = 0$

Multiply $x^{2}$ by $x$ .

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

$2 ((x - 3) (2 (x^{2} x) + 2 x (3 x) + 2 x \cdot 9 + 3 (- 3 x^{2} - 2 x - 6))) = 0$

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

$2 ((x - 3) (2 x^{2 + 1} + 2 x (3 x) + 2 x \cdot 9 + 3 (- 3 x^{2} - 2 x - 6))) = 0$

Add $2$ and $1$ .

$2 ((x - 3) (2 x^{3} + 2 x (3 x) + 2 x \cdot 9 + 3 (- 3 x^{2} - 2 x - 6))) = 0$

Rewrite using the commutative property of multiplication.

$2 ((x - 3) (2 x^{3} + 2 \cdot (3 x \cdot x) + 2 x \cdot 9 + 3 (- 3 x^{2} - 2 x - 6))) = 0$

Multiply $9$ by $2$ .

$2 ((x - 3) (2 x^{3} + 2 \cdot (3 x \cdot x) + 18 x + 3 (- 3 x^{2} - 2 x - 6))) = 0$

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$2 ((x - 3) (2 x^{3} + 2 \cdot (3 (x \cdot x)) + 18 x + 3 (- 3 x^{2} - 2 x - 6))) = 0$

Multiply $x$ by $x$ .

$2 ((x - 3) (2 x^{3} + 2 \cdot (3 x^{2}) + 18 x + 3 (- 3 x^{2} - 2 x - 6))) = 0$

Multiply $2$ by $3$ .

$2 ((x - 3) (2 x^{3} + 6 x^{2} + 18 x + 3 (- 3 x^{2} - 2 x - 6))) = 0$

Apply the distributive property.

$2 ((x - 3) (2 x^{3} + 6 x^{2} + 18 x + 3 (- 3 x^{2}) + 3 (- 2 x) + 3 \cdot - 6)) = 0$

Simplify.

Multiply $- 3$ by $3$ .

$2 ((x - 3) (2 x^{3} + 6 x^{2} + 18 x - 9 x^{2} + 3 (- 2 x) + 3 \cdot - 6)) = 0$

Multiply $- 2$ by $3$ .

$2 ((x - 3) (2 x^{3} + 6 x^{2} + 18 x - 9 x^{2} - 6 x + 3 \cdot - 6)) = 0$

Multiply $3$ by $- 6$ .

$2 ((x - 3) (2 x^{3} + 6 x^{2} + 18 x - 9 x^{2} - 6 x - 18)) = 0$

Subtract $9 x^{2}$ from $6 x^{2}$ .

$2 ((x - 3) (2 x^{3} - 3 x^{2} + 18 x - 6 x - 18)) = 0$

Subtract $6 x$ from $18 x$ .

$2 ((x - 3) (2 x^{3} - 3 x^{2} + 12 x - 18)) = 0$

Factor.

Rewrite $2 x^{3} - 3 x^{2} + 12 x - 18$ in a factored form.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 ((x - 3) ((2 x^{3} - 3 x^{2}) + 12 x - 18)) = 0$

Factor out the greatest common factor (GCF) from each group.

$2 ((x - 3) (x^{2} (2 x - 3) + 6 (2 x - 3))) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$2 ((x - 3) ((2 x - 3) (x^{2} + 6))) = 0$

Remove unnecessary parentheses.

$2 ((x - 3) (2 x - 3) (x^{2} + 6)) = 0$

Remove unnecessary parentheses.

$2 (x - 3) (2 x - 3) (x^{2} + 6) = 0$

Step 2

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

$x - 3 = 0$

$2 x - 3 = 0$

$x^{2} + 6 = 0$

Step 3

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 4

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Solve $2 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 5

Set $x^{2} + 6$ equal to $0$ and solve for $x$ .

Set $x^{2} + 6$ equal to $0$ .

$x^{2} + 6 = 0$

Solve $x^{2} + 6 = 0$ for $x$ .

Subtract $6$ from both sides of the equation.

$x^{2} = - 6$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 6}$

Simplify $\pm \sqrt{- 6}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$x = \pm \sqrt{- 1 (6)}$

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{6}$

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

$x = \pm i \sqrt{6}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{6}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{6}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{6}, - i \sqrt{6}$

Step 6

The final solution is all the values that make $2 (x - 3) (2 x - 3) (x^{2} + 6) = 0$ true.

$x = 3, \frac{3}{2}, i \sqrt{6}, - i \sqrt{6}$

Find All Complex Solutions 4x^4-18x^3+42x^2-108x+108=0

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