Calculus Examples

$x^{3} - 3 x^{2} + 4$

Step 1

Write $x^{3} - 3 x^{2} + 4$ as an equation.

$y = x^{3} - 3 x^{2} + 4$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = x^{3} - 3 x^{2} + 4$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $x^{3} - 3 x^{2} + 4 = 0$ .

$x^{3} - 3 x^{2} + 4 = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $x^{3} - 3 x^{2} + 4$ 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.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1$

Step 2.2.2.1.2

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

$\pm 1, \pm 4, \pm 2$

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.

${(- 1)}^{3} - 3 {(- 1)}^{2} + 4$

Step 2.2.2.1.3.2

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

$- 1 - 3 {(- 1)}^{2} + 4$

Step 2.2.2.1.3.3

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

$- 1 - 3 \cdot 1 + 4$

Step 2.2.2.1.3.4

Multiply $- 3$ by $1$ .

$- 1 - 3 + 4$

Step 2.2.2.1.3.5

Subtract $3$ from $- 1$ .

$- 4 + 4$

Step 2.2.2.1.3.6

Add $- 4$ and $4$ .

$0$

Step 2.2.2.1.4

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.

$\frac{x^{3} - 3 x^{2} + 4}{x + 1}$

Step 2.2.2.1.5

Divide $x^{3} - 3 x^{2} + 4$ by $x + 1$ .

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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 $0$ .

$x$

$1$

$x^{3}$

$3 x^{2}$

$0 x$

$4$

Step 2.2.2.1.5.2

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$

Step 2.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			+	$x^{3}$	+	$x^{2}$

Step 2.2.2.1.5.4

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$

Step 2.2.2.1.5.5

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$

Step 2.2.2.1.5.6

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$

Step 2.2.2.1.5.7

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$

Step 2.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	-	$4 x$

Step 2.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} - 4 x$

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$

Step 2.2.2.1.5.10

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

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$

Step 2.2.2.1.5.11

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

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$	+	$4$

Step 2.2.2.1.5.12

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	+	$4$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$	+	$4$

Step 2.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$4$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$	+	$4$
							+	$4 x$	+	$4$

Step 2.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x + 4$

				$x^{2}$	-	$4 x$	+	$4$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$	+	$4$
							-	$4 x$	-	$4$

Step 2.2.2.1.5.15

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

				$x^{2}$	-	$4 x$	+	$4$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$	+	$4$
							-	$4 x$	-	$4$
										$0$

Step 2.2.2.1.5.16

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

$x^{2} - 4 x + 4$

Step 2.2.2.1.6

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

$(x + 1) (x^{2} - 4 x + 4) = 0$

Step 2.2.2.2

Factor using the perfect square rule.

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Step 2.2.2.2.1

Rewrite $4$ as $2^{2}$ .

$(x + 1) (x^{2} - 4 x + 2^{2}) = 0$

Step 2.2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 2.2.2.2.3

Rewrite the polynomial.

$(x + 1) (x^{2} - 2 \cdot x \cdot 2 + 2^{2}) = 0$

Step 2.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 2.2.3

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

$x + 1 = 0$

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

Step 2.2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.2.5

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set ${(x - 2)}^{2}$ equal to $0$ .

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

Step 2.2.5.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 2.2.5.2.1

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

$x - 2 = 0$

Step 2.2.5.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.2.6

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

$x = - 1, 2$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(- 1, 0), (2, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = {(0)}^{3} - 3 {(0)}^{2} + 4$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = 0^{3} - 3 {(0)}^{2} + 4$

Step 3.2.2

Remove parentheses.

$y = 0^{3} - 3 \cdot 0^{2} + 4$

Step 3.2.3

Remove parentheses.

$y = {(0)}^{3} - 3 {(0)}^{2} + 4$

Step 3.2.4

Simplify ${(0)}^{3} - 3 {(0)}^{2} + 4$ .

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 - 3 {(0)}^{2} + 4$

Step 3.2.4.1.2

Raising $0$ to any positive power yields $0$ .

$y = 0 - 3 \cdot 0 + 4$

Step 3.2.4.1.3

Multiply $- 3$ by $0$ .

$y = 0 + 0 + 4$

Step 3.2.4.2

Simplify by adding numbers.

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Step 3.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 4$

Step 3.2.4.2.2

Add $0$ and $4$ .

$y = 4$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 4)$

Step 4

List the intersections.

x-intercept(s): $(- 1, 0), (2, 0)$

y-intercept(s): $(0, 4)$

Step 5