Solve by Factoring x^3-10x^2+27x-18=0

$x^{3} - 10 x^{2} + 27 x - 18 = 0$

Step 1

Factor the left side of the equation.

Factor $x^{3} - 10 x^{2} + 27 x - 18$ 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.

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

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

$\pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6$

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.

$1^{3} - 10 \cdot 1^{2} + 27 \cdot 1 - 18$

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

$1 - 10 \cdot 1^{2} + 27 \cdot 1 - 18$

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

$1 - 10 \cdot 1 + 27 \cdot 1 - 18$

Multiply $- 10$ by $1$ .

$1 - 10 + 27 \cdot 1 - 18$

Subtract $10$ from $1$ .

$- 9 + 27 \cdot 1 - 18$

Multiply $27$ by $1$ .

$- 9 + 27 - 18$

Add $- 9$ and $27$ .

$18 - 18$

Subtract $18$ from $18$ .

$0$

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.

$\frac{x^{3} - 10 x^{2} + 27 x - 18}{x - 1}$

Divide $x^{3} - 10 x^{2} + 27 x - 18$ 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 $0$ .

$x$

$1$

$x^{3}$

$10 x^{2}$

$27 x$

$18$

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			+	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$27 x$

Divide the highest order term in the dividend $- 9 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$9 x$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$27 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$27 x$
					-	$9 x^{2}$	+	$9 x$

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

				$x^{2}$	-	$9 x$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$27 x$
					+	$9 x^{2}$	-	$9 x$

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

				$x^{2}$	-	$9 x$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$27 x$
					+	$9 x^{2}$	-	$9 x$
							+	$18 x$

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

				$x^{2}$	-	$9 x$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$27 x$
					+	$9 x^{2}$	-	$9 x$
							+	$18 x$	-	$18$

Divide the highest order term in the dividend $18 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$9 x$	+	$18$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$27 x$
					+	$9 x^{2}$	-	$9 x$
							+	$18 x$	-	$18$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$	+	$18$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$27 x$
					+	$9 x^{2}$	-	$9 x$
							+	$18 x$	-	$18$
							+	$18 x$	-	$18$

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

				$x^{2}$	-	$9 x$	+	$18$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$27 x$
					+	$9 x^{2}$	-	$9 x$
							+	$18 x$	-	$18$
							-	$18 x$	+	$18$

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

				$x^{2}$	-	$9 x$	+	$18$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$27 x$	-	$18$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$27 x$
					+	$9 x^{2}$	-	$9 x$
							+	$18 x$	-	$18$
							-	$18 x$	+	$18$
										$0$

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

$x^{2} - 9 x + 18$

Write $x^{3} - 10 x^{2} + 27 x - 18$ as a set of factors.

$(x - 1) (x^{2} - 9 x + 18) = 0$

Factor $x^{2} - 9 x + 18$ 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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$(x - 1) ((x - 6) (x - 3)) = 0$

Remove unnecessary parentheses.

$(x - 1) (x - 6) (x - 3) = 0$

Step 2

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

$x - 1 = 0$

$x - 6 = 0$

$x - 3 = 0$

Step 3

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 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 $x - 3$ 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 6

The final solution is all the values that make $(x - 1) (x - 6) (x - 3) = 0$ true.

$x = 1, 6, 3$

Solve by Factoring x^3-10x^2+27x-18=0

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