Finite Math Examples

$2 p - 2 \sqrt{4 p^{3} + 16 p^{2} + 28 p - 48}$

Step 1

Set the radicand in $\sqrt{4 p^{3} + 16 p^{2} + 28 p - 48}$ greater than or equal to $0$ to find where the expression is defined.

$4 p^{3} + 16 p^{2} + 28 p - 48 \geq 0$

Step 2

Solve for $p$ .

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Step 2.1

Convert the inequality to an equation.

$4 p^{3} + 16 p^{2} + 28 p - 48 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $4$ out of $4 p^{3} + 16 p^{2} + 28 p - 48$ .

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Step 2.2.1.1

Factor $4$ out of $4 p^{3}$ .

$4 (p^{3}) + 16 p^{2} + 28 p - 48 = 0$

Step 2.2.1.2

Factor $4$ out of $16 p^{2}$ .

$4 (p^{3}) + 4 (4 p^{2}) + 28 p - 48 = 0$

Step 2.2.1.3

Factor $4$ out of $28 p$ .

$4 (p^{3}) + 4 (4 p^{2}) + 4 (7 p) - 48 = 0$

Step 2.2.1.4

Factor $4$ out of $- 48$ .

$4 p^{3} + 4 (4 p^{2}) + 4 (7 p) + 4 \cdot - 12 = 0$

Step 2.2.1.5

Factor $4$ out of $4 p^{3} + 4 (4 p^{2})$ .

$4 (p^{3} + 4 p^{2}) + 4 (7 p) + 4 \cdot - 12 = 0$

Step 2.2.1.6

Factor $4$ out of $4 (p^{3} + 4 p^{2}) + 4 (7 p)$ .

$4 (p^{3} + 4 p^{2} + 7 p) + 4 \cdot - 12 = 0$

Step 2.2.1.7

Factor $4$ out of $4 (p^{3} + 4 p^{2} + 7 p) + 4 \cdot - 12$ .

$4 (p^{3} + 4 p^{2} + 7 p - 12) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $p^{3} + 4 p^{2} + 7 p - 12$ using the rational roots test.

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Step 2.2.2.1.1

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.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1$

Step 2.2.2.1.2

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

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

Step 2.2.2.1.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 2.2.2.1.3.1

Substitute $1$ into the polynomial.

$1^{3} + 4 \cdot 1^{2} + 7 \cdot 1 - 12$

Step 2.2.2.1.3.2

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

$1 + 4 \cdot 1^{2} + 7 \cdot 1 - 12$

Step 2.2.2.1.3.3

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

$1 + 4 \cdot 1 + 7 \cdot 1 - 12$

Step 2.2.2.1.3.4

Multiply $4$ by $1$ .

$1 + 4 + 7 \cdot 1 - 12$

Step 2.2.2.1.3.5

Add $1$ and $4$ .

$5 + 7 \cdot 1 - 12$

Step 2.2.2.1.3.6

Multiply $7$ by $1$ .

$5 + 7 - 12$

Step 2.2.2.1.3.7

Add $5$ and $7$ .

$12 - 12$

Step 2.2.2.1.3.8

Subtract $12$ from $12$ .

$0$

Step 2.2.2.1.4

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

$\frac{p^{3} + 4 p^{2} + 7 p - 12}{p - 1}$

Step 2.2.2.1.5

Divide $p^{3} + 4 p^{2} + 7 p - 12$ by $p - 1$ .

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Step 2.2.2.1.5.1

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

$p$

$1$

$p^{3}$

$4 p^{2}$

$7 p$

$12$

Step 2.2.2.1.5.2

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

					$p^{2}$
	$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$

Step 2.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$p^{2}$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			+	$p^{3}$	-	$p^{2}$

Step 2.2.2.1.5.4

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

				$p^{2}$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$

Step 2.2.2.1.5.5

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

				$p^{2}$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$

Step 2.2.2.1.5.6

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

				$p^{2}$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$	+	$7 p$

Step 2.2.2.1.5.7

Divide the highest order term in the dividend $5 p^{2}$ by the highest order term in divisor $p$ .

				$p^{2}$	+	$5 p$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$	+	$7 p$

Step 2.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$p^{2}$	+	$5 p$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$	+	$7 p$
					+	$5 p^{2}$	-	$5 p$

Step 2.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $5 p^{2} - 5 p$

				$p^{2}$	+	$5 p$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$	+	$7 p$
					-	$5 p^{2}$	+	$5 p$

Step 2.2.2.1.5.10

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

				$p^{2}$	+	$5 p$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$	+	$7 p$
					-	$5 p^{2}$	+	$5 p$
							+	$12 p$

Step 2.2.2.1.5.11

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

				$p^{2}$	+	$5 p$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$	+	$7 p$
					-	$5 p^{2}$	+	$5 p$
							+	$12 p$	-	$12$

Step 2.2.2.1.5.12

Divide the highest order term in the dividend $12 p$ by the highest order term in divisor $p$ .

				$p^{2}$	+	$5 p$	+	$12$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$	+	$7 p$
					-	$5 p^{2}$	+	$5 p$
							+	$12 p$	-	$12$

Step 2.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$p^{2}$	+	$5 p$	+	$12$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$	+	$7 p$
					-	$5 p^{2}$	+	$5 p$
							+	$12 p$	-	$12$
							+	$12 p$	-	$12$

Step 2.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $12 p - 12$

				$p^{2}$	+	$5 p$	+	$12$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$	+	$7 p$
					-	$5 p^{2}$	+	$5 p$
							+	$12 p$	-	$12$
							-	$12 p$	+	$12$

Step 2.2.2.1.5.15

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

				$p^{2}$	+	$5 p$	+	$12$
$p$	-	$1$		$p^{3}$	+	$4 p^{2}$	+	$7 p$	-	$12$
			-	$p^{3}$	+	$p^{2}$
					+	$5 p^{2}$	+	$7 p$
					-	$5 p^{2}$	+	$5 p$
							+	$12 p$	-	$12$
							-	$12 p$	+	$12$
										$0$

Step 2.2.2.1.5.16

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

$p^{2} + 5 p + 12$

Step 2.2.2.1.6

Write $p^{3} + 4 p^{2} + 7 p - 12$ as a set of factors.

$4 ((p - 1) (p^{2} + 5 p + 12)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$4 (p - 1) (p^{2} + 5 p + 12) = 0$

Step 2.3

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

$p - 1 = 0$

$p^{2} + 5 p + 12 = 0$

Step 2.4

Set $p - 1$ equal to $0$ and solve for $p$ .

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Step 2.4.1

Set $p - 1$ equal to $0$ .

$p - 1 = 0$

Step 2.4.2

Add $1$ to both sides of the equation.

$p = 1$

Step 2.5

Set $p^{2} + 5 p + 12$ equal to $0$ and solve for $p$ .

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Step 2.5.1

Set $p^{2} + 5 p + 12$ equal to $0$ .

$p^{2} + 5 p + 12 = 0$

Step 2.5.2

Solve $p^{2} + 5 p + 12 = 0$ for $p$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5.2.2

Substitute the values $a = 1$ , $b = 5$ , and $c = 12$ into the quadratic formula and solve for $p$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot 12)}}{2 \cdot 1}$

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

Raise $5$ to the power of $2$ .

$p = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 12$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$p = \frac{- 5 \pm \sqrt{25 - 4 \cdot 12}}{2 \cdot 1}$

Step 2.5.2.3.1.2.2

Multiply $- 4$ by $12$ .

$p = \frac{- 5 \pm \sqrt{25 - 48}}{2 \cdot 1}$

Step 2.5.2.3.1.3

Subtract $48$ from $25$ .

$p = \frac{- 5 \pm \sqrt{- 23}}{2 \cdot 1}$

Step 2.5.2.3.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$p = \frac{- 5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 1}$

Step 2.5.2.3.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$p = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 1}$

Step 2.5.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$p = \frac{- 5 \pm i \sqrt{23}}{2 \cdot 1}$

Step 2.5.2.3.2

Multiply $2$ by $1$ .

$p = \frac{- 5 \pm i \sqrt{23}}{2}$

Step 2.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.5.2.4.1

Simplify the numerator.

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Step 2.5.2.4.1.1

Raise $5$ to the power of $2$ .

$p = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Step 2.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 12$ .

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Step 2.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$p = \frac{- 5 \pm \sqrt{25 - 4 \cdot 12}}{2 \cdot 1}$

Step 2.5.2.4.1.2.2

Multiply $- 4$ by $12$ .

$p = \frac{- 5 \pm \sqrt{25 - 48}}{2 \cdot 1}$

Step 2.5.2.4.1.3

Subtract $48$ from $25$ .

$p = \frac{- 5 \pm \sqrt{- 23}}{2 \cdot 1}$

Step 2.5.2.4.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$p = \frac{- 5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 1}$

Step 2.5.2.4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$p = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 1}$

Step 2.5.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$p = \frac{- 5 \pm i \sqrt{23}}{2 \cdot 1}$

Step 2.5.2.4.2

Multiply $2$ by $1$ .

$p = \frac{- 5 \pm i \sqrt{23}}{2}$

Step 2.5.2.4.3

Change the $\pm$ to $+$ .

$p = \frac{- 5 + i \sqrt{23}}{2}$

Step 2.5.2.4.4

Rewrite $- 5$ as $- 1 (5)$ .

$p = \frac{- 1 \cdot 5 + i \sqrt{23}}{2}$

Step 2.5.2.4.5

Factor $- 1$ out of $i \sqrt{23}$ .

$p = \frac{- 1 \cdot 5 - (- i \sqrt{23})}{2}$

Step 2.5.2.4.6

Factor $- 1$ out of $- 1 (5) - (- i \sqrt{23})$ .

$p = \frac{- 1 (5 - i \sqrt{23})}{2}$

Step 2.5.2.4.7

Move the negative in front of the fraction.

$p = - \frac{5 - i \sqrt{23}}{2}$

Step 2.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.5.2.5.1

Simplify the numerator.

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Step 2.5.2.5.1.1

Raise $5$ to the power of $2$ .

$p = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Step 2.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 12$ .

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Step 2.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$p = \frac{- 5 \pm \sqrt{25 - 4 \cdot 12}}{2 \cdot 1}$

Step 2.5.2.5.1.2.2

Multiply $- 4$ by $12$ .

$p = \frac{- 5 \pm \sqrt{25 - 48}}{2 \cdot 1}$

Step 2.5.2.5.1.3

Subtract $48$ from $25$ .

$p = \frac{- 5 \pm \sqrt{- 23}}{2 \cdot 1}$

Step 2.5.2.5.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$p = \frac{- 5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 1}$

Step 2.5.2.5.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$p = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 1}$

Step 2.5.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$p = \frac{- 5 \pm i \sqrt{23}}{2 \cdot 1}$

Step 2.5.2.5.2

Multiply $2$ by $1$ .

$p = \frac{- 5 \pm i \sqrt{23}}{2}$

Step 2.5.2.5.3

Change the $\pm$ to $-$ .

$p = \frac{- 5 - i \sqrt{23}}{2}$

Step 2.5.2.5.4

Rewrite $- 5$ as $- 1 (5)$ .

$p = \frac{- 1 \cdot 5 - i \sqrt{23}}{2}$

Step 2.5.2.5.5

Factor $- 1$ out of $- i \sqrt{23}$ .

$p = \frac{- 1 \cdot 5 - (i \sqrt{23})}{2}$

Step 2.5.2.5.6

Factor $- 1$ out of $- 1 (5) - (i \sqrt{23})$ .

$p = \frac{- 1 (5 + i \sqrt{23})}{2}$

Step 2.5.2.5.7

Move the negative in front of the fraction.

$p = - \frac{5 + i \sqrt{23}}{2}$

Step 2.5.2.6

The final answer is the combination of both solutions.

$p = - \frac{5 - i \sqrt{23}}{2}, - \frac{5 + i \sqrt{23}}{2}$

Step 2.6

The final solution is all the values that make $4 (p - 1) (p^{2} + 5 p + 12) = 0$ true.

$p = 1, - \frac{5 - i \sqrt{23}}{2}, - \frac{5 + i \sqrt{23}}{2}$

Step 2.7

Identify the leading coefficient.

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Step 2.7.1

The leading term in a polynomial is the term with the highest degree.

$4 p^{3}$

Step 2.7.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$4$

Step 2.8

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $4 p^{3} + 16 p^{2} + 28 p - 48$ is always greater than $0$ .

All real numbers

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${p | p \in ℝ}$

Step 4