Find All Complex Solutions x^3-10x^2+19x+30=0

$x^{3} - 10 x^{2} + 19 x + 30 = 0$

Step 1

Factor the left side of the equation.

Factor $x^{3} - 10 x^{2} + 19 x + 30$ 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.

$p = \pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6 q = \pm 1$

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

$\pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

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.

${(- 1)}^{3} - 10 {(- 1)}^{2} + 19 \cdot - 1 + 30$

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

$- 1 - 10 {(- 1)}^{2} + 19 \cdot - 1 + 30$

Raise $- 1$ to the power of $2$ .

$- 1 - 10 \cdot 1 + 19 \cdot - 1 + 30$

Multiply $- 10$ by $1$ .

$- 1 - 10 + 19 \cdot - 1 + 30$

Subtract $10$ from $- 1$ .

$- 11 + 19 \cdot - 1 + 30$

Multiply $19$ by $- 1$ .

$- 11 - 19 + 30$

Subtract $19$ from $- 11$ .

$- 30 + 30$

Add $- 30$ and $30$ .

$0$

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.

$\frac{x^{3} - 10 x^{2} + 19 x + 30}{x + 1}$

Divide $x^{3} - 10 x^{2} + 19 x + 30$ 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 $0$ .

$x$

$1$

$x^{3}$

$10 x^{2}$

$19 x$

$30$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$	+	$19 x$

Divide the highest order term in the dividend $- 11 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$11 x$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$	+	$19 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$11 x$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$	+	$19 x$
					-	$11 x^{2}$	-	$11 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 11 x^{2} - 11 x$

				$x^{2}$	-	$11 x$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$	+	$19 x$
					+	$11 x^{2}$	+	$11 x$

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

				$x^{2}$	-	$11 x$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$	+	$19 x$
					+	$11 x^{2}$	+	$11 x$
							+	$30 x$

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

				$x^{2}$	-	$11 x$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$	+	$19 x$
					+	$11 x^{2}$	+	$11 x$
							+	$30 x$	+	$30$

Divide the highest order term in the dividend $30 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$11 x$	+	$30$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$	+	$19 x$
					+	$11 x^{2}$	+	$11 x$
							+	$30 x$	+	$30$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$11 x$	+	$30$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$	+	$19 x$
					+	$11 x^{2}$	+	$11 x$
							+	$30 x$	+	$30$
							+	$30 x$	+	$30$

The expression needs to be subtracted from the dividend, so change all the signs in $30 x + 30$

				$x^{2}$	-	$11 x$	+	$30$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$	+	$19 x$
					+	$11 x^{2}$	+	$11 x$
							+	$30 x$	+	$30$
							-	$30 x$	-	$30$

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

				$x^{2}$	-	$11 x$	+	$30$
$x$	+	$1$		$x^{3}$	-	$10 x^{2}$	+	$19 x$	+	$30$
			-	$x^{3}$	-	$x^{2}$
					-	$11 x^{2}$	+	$19 x$
					+	$11 x^{2}$	+	$11 x$
							+	$30 x$	+	$30$
							-	$30 x$	-	$30$
										$0$

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

$x^{2} - 11 x + 30$

Write $x^{3} - 10 x^{2} + 19 x + 30$ as a set of factors.

$(x + 1) (x^{2} - 11 x + 30) = 0$

Factor $x^{2} - 11 x + 30$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $30$ and whose sum is $- 11$ .

$- 6, - 5$

Write the factored form using these integers.

$(x + 1) ((x - 6) (x - 5)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x - 6) (x - 5) = 0$

Step 2

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

$x + 1 = 0$

$x - 6 = 0$

$x - 5 = 0$

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 $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 5

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 6

The final solution is all the values that make $(x + 1) (x - 6) (x - 5) = 0$ true.

$x = - 1, 6, 5$

Find All Complex Solutions x^3-10x^2+19x+30=0

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