Find All Complex Solutions x^3-2x^2=19x-20

$x^{3} - 2 x^{2} = 19 x - 20$

Step 1

Move all the expressions to the left side of the equation.

Subtract $19 x$ from both sides of the equation.

$x^{3} - 2 x^{2} - 19 x = - 20$

Add $20$ to both sides of the equation.

$x^{3} - 2 x^{2} - 19 x + 20 = 0$

Step 2

Factor the left side of the equation.

Factor $x^{3} - 2 x^{2} - 19 x + 20$ 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.

$p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5 q = \pm 1$

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

$\pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

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.

$1^{3} - 2 \cdot 1^{2} - 19 \cdot 1 + 20$

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

$1 - 2 \cdot 1^{2} - 19 \cdot 1 + 20$

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

$1 - 2 \cdot 1 - 19 \cdot 1 + 20$

Multiply $- 2$ by $1$ .

$1 - 2 - 19 \cdot 1 + 20$

Subtract $2$ from $1$ .

$- 1 - 19 \cdot 1 + 20$

Multiply $- 19$ by $1$ .

$- 1 - 19 + 20$

Subtract $19$ from $- 1$ .

$- 20 + 20$

Add $- 20$ and $20$ .

$0$

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.

$\frac{x^{3} - 2 x^{2} - 19 x + 20}{x - 1}$

Divide $x^{3} - 2 x^{2} - 19 x + 20$ 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 $0$ .

$x$

$1$

$x^{3}$

$2 x^{2}$

$19 x$

$20$

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	-	$19 x$

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	-	$19 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	-	$19 x$
					-	$x^{2}$	+	$x$

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	-	$19 x$
					+	$x^{2}$	-	$x$

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	-	$19 x$
					+	$x^{2}$	-	$x$
							-	$20 x$

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

				$x^{2}$	-	$x$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	-	$19 x$
					+	$x^{2}$	-	$x$
							-	$20 x$	+	$20$

Divide the highest order term in the dividend $- 20 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$x$	-	$20$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	-	$19 x$
					+	$x^{2}$	-	$x$
							-	$20 x$	+	$20$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$20$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	-	$19 x$
					+	$x^{2}$	-	$x$
							-	$20 x$	+	$20$
							-	$20 x$	+	$20$

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x + 20$

				$x^{2}$	-	$x$	-	$20$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	-	$19 x$
					+	$x^{2}$	-	$x$
							-	$20 x$	+	$20$
							+	$20 x$	-	$20$

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

				$x^{2}$	-	$x$	-	$20$
$x$	-	$1$		$x^{3}$	-	$2 x^{2}$	-	$19 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$
					-	$x^{2}$	-	$19 x$
					+	$x^{2}$	-	$x$
							-	$20 x$	+	$20$
							+	$20 x$	-	$20$
										$0$

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

$x^{2} - x - 20$

Write $x^{3} - 2 x^{2} - 19 x + 20$ as a set of factors.

$(x - 1) (x^{2} - x - 20) = 0$

Factor $x^{2} - x - 20$ 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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Write the factored form using these integers.

$(x - 1) ((x - 5) (x + 4)) = 0$

Remove unnecessary parentheses.

$(x - 1) (x - 5) (x + 4) = 0$

Step 3

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

$x - 1 = 0$

$x - 5 = 0$

$x + 4 = 0$

Step 4

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 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

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

The final solution is all the values that make $(x - 1) (x - 5) (x + 4) = 0$ true.

$x = 1, 5, - 4$

Find All Complex Solutions x^3-2x^2=19x-20

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