Solve the Inequality for x x^3-10x^2+23x-14<0

$x^{3} - 10 x^{2} + 23 x - 14 < 0$

Convert the inequality to an equation.

$x^{3} - 10 x^{2} + 23 x - 14 = 0$

Factor the left side of the equation.

Factor $x^{3} - 10 x^{2} + 23 x - 14$ 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.

$p = \pm 1, \pm 14, \pm 2, \pm 7 q = \pm 1$

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

$\pm 1, \pm 14, \pm 2, \pm 7$

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.

$1^{3} - 10 \cdot 1^{2} + 23 \cdot 1 - 14$

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

$1 - 10 \cdot 1^{2} + 23 \cdot 1 - 14$

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

$1 - 10 \cdot 1 + 23 \cdot 1 - 14$

Multiply $- 10$ by $1$ .

$1 - 10 + 23 \cdot 1 - 14$

Subtract $10$ from $1$ .

$- 9 + 23 \cdot 1 - 14$

Multiply $23$ by $1$ .

$- 9 + 23 - 14$

Add $- 9$ and $23$ .

$14 - 14$

Subtract $14$ from $14$ .

$0$

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.

$\frac{x^{3} - 10 x^{2} + 23 x - 14}{x - 1}$

Divide $x^{3} - 10 x^{2} + 23 x - 14$ 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 $0$ .

$x$

$1$

$x^{3}$

$10 x^{2}$

$23 x$

$14$

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			+	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$23 x$

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

				$x^{2}$	-	$9 x$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$23 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$23 x$
					-	$9 x^{2}$	+	$9 x$

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

				$x^{2}$	-	$9 x$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$23 x$
					+	$9 x^{2}$	-	$9 x$

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

				$x^{2}$	-	$9 x$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$23 x$
					+	$9 x^{2}$	-	$9 x$
							+	$14 x$

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

				$x^{2}$	-	$9 x$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$23 x$
					+	$9 x^{2}$	-	$9 x$
							+	$14 x$	-	$14$

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

				$x^{2}$	-	$9 x$	+	$14$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$23 x$
					+	$9 x^{2}$	-	$9 x$
							+	$14 x$	-	$14$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$	+	$14$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$23 x$
					+	$9 x^{2}$	-	$9 x$
							+	$14 x$	-	$14$
							+	$14 x$	-	$14$

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

				$x^{2}$	-	$9 x$	+	$14$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$23 x$
					+	$9 x^{2}$	-	$9 x$
							+	$14 x$	-	$14$
							-	$14 x$	+	$14$

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

				$x^{2}$	-	$9 x$	+	$14$
$x$	-	$1$		$x^{3}$	-	$10 x^{2}$	+	$23 x$	-	$14$
			-	$x^{3}$	+	$x^{2}$
					-	$9 x^{2}$	+	$23 x$
					+	$9 x^{2}$	-	$9 x$
							+	$14 x$	-	$14$
							-	$14 x$	+	$14$
										$0$

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

$x^{2} - 9 x + 14$

Write $x^{3} - 10 x^{2} + 23 x - 14$ as a set of factors.

$(x - 1) (x^{2} - 9 x + 14) = 0$

Factor $x^{2} - 9 x + 14$ 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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

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

$x - 1 = 0$

$x - 7 = 0$

$x - 2 = 0$

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.

$x = 1$

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.

$x = 7$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

$x = 1, 7, 2$

Use each root to create test intervals.

$x < 1$

$1 < x < 2$

$2 < x < 7$

$x > 7$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < 1$ to see if it makes the inequality true.

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

${(0)}^{3} - 10 {(0)}^{2} + 23 (0) - 14 < 0$

The left side $- 14$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $1 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 2$ and see if this value makes the original inequality true.

$x = 1.5$

Replace $x$ with $1.5$ in the original inequality.

${(1.5)}^{3} - 10 {(1.5)}^{2} + 23 (1.5) - 14 < 0$

The left side $1.375$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $2 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 7$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

${(4)}^{3} - 10 {(4)}^{2} + 23 (4) - 14 < 0$

The left side $- 18$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 7$ to see if it makes the inequality true.

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

${(10)}^{3} - 10 {(10)}^{2} + 23 (10) - 14 < 0$

The left side $216$ is not less than the right side $0$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < 1$ True

$1 < x < 2$ False

$2 < x < 7$ True

$x > 7$ False

$x < 1$ True

$1 < x < 2$ False

$2 < x < 7$ True

$x > 7$ False

The solution consists of all of the true intervals.

$x < 1$ or $2 < x < 7$

The result can be shown in multiple forms.

Inequality Form:

$x < 1 or 2 < x < 7$

Interval Notation:

$(- \infty, 1) \cup (2, 7)$

Solve the Inequality for x x^3-10x^2+23x-14<0

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