Find All Complex Solutions 6x^3-47x^2+36x-7=0

$6 x^{3} - 47 x^{2} + 36 x - 7 = 0$

Step 1

Factor the left side of the equation.

Factor $6 x^{3} - 47 x^{2} + 36 x - 7$ 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.

$p = \pm 1, \pm 7 q = \pm 1, \pm 6, \pm 2, \pm 3$

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

$\pm 1, \pm 0.1\overline{6}, \pm 0.5, \pm 0.\overline{3}, \pm 7, \pm 1.1\overline{6}, \pm 3.5, \pm 2.\overline{3}$

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

Substitute $0.5$ into the polynomial.

$6 \cdot {0.5}^{3} - 47 \cdot {0.5}^{2} + 36 \cdot 0.5 - 7$

Raise $0.5$ to the power of $3$ .

$6 \cdot 0.125 - 47 \cdot {0.5}^{2} + 36 \cdot 0.5 - 7$

Multiply $6$ by $0.125$ .

$0.75 - 47 \cdot {0.5}^{2} + 36 \cdot 0.5 - 7$

Raise $0.5$ to the power of $2$ .

$0.75 - 47 \cdot 0.25 + 36 \cdot 0.5 - 7$

Multiply $- 47$ by $0.25$ .

$0.75 - 11.75 + 36 \cdot 0.5 - 7$

Subtract $11.75$ from $0.75$ .

$- 11 + 36 \cdot 0.5 - 7$

Multiply $36$ by $0.5$ .

$- 11 + 18 - 7$

Add $- 11$ and $18$ .

$7 - 7$

Subtract $7$ from $7$ .

$0$

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{6 x^{3} - 47 x^{2} + 36 x - 7}{2 x - 1}$

Divide $6 x^{3} - 47 x^{2} + 36 x - 7$ by $2 x - 1$ .

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

$2 x$

$1$

$6 x^{3}$

$47 x^{2}$

$36 x$

$7$

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $2 x$ .

					$3 x^{2}$
	$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			+	$6 x^{3}$	-	$3 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} - 3 x^{2}$

				$3 x^{2}$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$

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

				$3 x^{2}$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$

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

				$3 x^{2}$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$	+	$36 x$

Divide the highest order term in the dividend $- 44 x^{2}$ by the highest order term in divisor $2 x$ .

				$3 x^{2}$	-	$22 x$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$	+	$36 x$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$22 x$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$	+	$36 x$
					-	$44 x^{2}$	+	$22 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 44 x^{2} + 22 x$

				$3 x^{2}$	-	$22 x$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$	+	$36 x$
					+	$44 x^{2}$	-	$22 x$

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

				$3 x^{2}$	-	$22 x$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$	+	$36 x$
					+	$44 x^{2}$	-	$22 x$
							+	$14 x$

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

				$3 x^{2}$	-	$22 x$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$	+	$36 x$
					+	$44 x^{2}$	-	$22 x$
							+	$14 x$	-	$7$

Divide the highest order term in the dividend $14 x$ by the highest order term in divisor $2 x$ .

				$3 x^{2}$	-	$22 x$	+	$7$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$	+	$36 x$
					+	$44 x^{2}$	-	$22 x$
							+	$14 x$	-	$7$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$22 x$	+	$7$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$	+	$36 x$
					+	$44 x^{2}$	-	$22 x$
							+	$14 x$	-	$7$
							+	$14 x$	-	$7$

The expression needs to be subtracted from the dividend, so change all the signs in $14 x - 7$

				$3 x^{2}$	-	$22 x$	+	$7$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$	+	$36 x$
					+	$44 x^{2}$	-	$22 x$
							+	$14 x$	-	$7$
							-	$14 x$	+	$7$

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

				$3 x^{2}$	-	$22 x$	+	$7$
$2 x$	-	$1$		$6 x^{3}$	-	$47 x^{2}$	+	$36 x$	-	$7$
			-	$6 x^{3}$	+	$3 x^{2}$
					-	$44 x^{2}$	+	$36 x$
					+	$44 x^{2}$	-	$22 x$
							+	$14 x$	-	$7$
							-	$14 x$	+	$7$
										$0$

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

$3 x^{2} - 22 x + 7$

Write $6 x^{3} - 47 x^{2} + 36 x - 7$ as a set of factors.

$(2 x - 1) (3 x^{2} - 22 x + 7) = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 7 = 21$ and whose sum is $b = - 22$ .

Factor $- 22$ out of $- 22 x$ .

$(2 x - 1) (3 x^{2} - 22 x + 7) = 0$

Rewrite $- 22$ as $- 1$ plus $- 21$

$(2 x - 1) (3 x^{2} + (- 1 - 21) x + 7) = 0$

Apply the distributive property.

$(2 x - 1) (3 x^{2} - 1 x - 21 x + 7) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x - 1) ((3 x^{2} - 1 x) - 21 x + 7) = 0$

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

$(2 x - 1) (x (3 x - 1) - 7 (3 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(2 x - 1) ((3 x - 1) (x - 7)) = 0$

Remove unnecessary parentheses.

$(2 x - 1) (3 x - 1) (x - 7) = 0$

Step 2

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

$2 x - 1 = 0$

$3 x - 1 = 0$

$x - 7 = 0$

Step 3

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 4

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Solve $3 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 5

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 6

The final solution is all the values that make $(2 x - 1) (3 x - 1) (x - 7) = 0$ true.

$x = \frac{1}{2}, \frac{1}{3}, 7$

Find All Complex Solutions 6x^3-47x^2+36x-7=0

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