Calculus Examples

$2 x^{3} - 7 x^{2} + x + 10 = y$

Step 1

Rewrite the equation as $y = 2 x^{3} - 7 x^{2} + x + 10$ .

$y = 2 x^{3} - 7 x^{2} + x + 10$

Step 2

Set $y$ equal to $0$ .

$2 x^{3} - 7 x^{2} + x + 10 = 0$

Step 3

Solve for $x$ .

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Step 3.1

Factor the left side of the equation.

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Step 3.1.1

Factor $2 x^{3} - 7 x^{2} + x + 10$ using the rational roots test.

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Step 3.1.1.1

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.

$p = \pm 1, \pm 10, \pm 2, \pm 5$

$q = \pm 1, \pm 2$

Step 3.1.1.2

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

$\pm 1, \pm 0.5, \pm 10, \pm 5, \pm 2, \pm 2.5$

Step 3.1.1.3

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 3.1.1.3.1

Substitute $- 1$ into the polynomial.

$2 {(- 1)}^{3} - 7 {(- 1)}^{2} - 1 + 10$

Step 3.1.1.3.2

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

$2 \cdot - 1 - 7 {(- 1)}^{2} - 1 + 10$

Step 3.1.1.3.3

Multiply $2$ by $- 1$ .

$- 2 - 7 {(- 1)}^{2} - 1 + 10$

Step 3.1.1.3.4

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

$- 2 - 7 \cdot 1 - 1 + 10$

Step 3.1.1.3.5

Multiply $- 7$ by $1$ .

$- 2 - 7 - 1 + 10$

Step 3.1.1.3.6

Subtract $7$ from $- 2$ .

$- 9 - 1 + 10$

Step 3.1.1.3.7

Subtract $1$ from $- 9$ .

$- 10 + 10$

Step 3.1.1.3.8

Add $- 10$ and $10$ .

$0$

Step 3.1.1.4

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.

$\frac{2 x^{3} - 7 x^{2} + x + 10}{x + 1}$

Step 3.1.1.5

Divide $2 x^{3} - 7 x^{2} + x + 10$ by $x + 1$ .

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Step 3.1.1.5.1

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

$x$

$1$

$2 x^{3}$

$7 x^{2}$

$x$

$10$

Step 3.1.1.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

					$2 x^{2}$
	$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$

Step 3.1.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			+	$2 x^{3}$	+	$2 x^{2}$

Step 3.1.1.5.4

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$

Step 3.1.1.5.5

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$

Step 3.1.1.5.6

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$x$

Step 3.1.1.5.7

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

				$2 x^{2}$	-	$9 x$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$x$

Step 3.1.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$9 x$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$x$
					-	$9 x^{2}$	-	$9 x$

Step 3.1.1.5.9

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

				$2 x^{2}$	-	$9 x$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$x$
					+	$9 x^{2}$	+	$9 x$

Step 3.1.1.5.10

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

				$2 x^{2}$	-	$9 x$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$x$
					+	$9 x^{2}$	+	$9 x$
							+	$10 x$

Step 3.1.1.5.11

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

				$2 x^{2}$	-	$9 x$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$x$
					+	$9 x^{2}$	+	$9 x$
							+	$10 x$	+	$10$

Step 3.1.1.5.12

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$9 x$	+	$10$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$x$
					+	$9 x^{2}$	+	$9 x$
							+	$10 x$	+	$10$

Step 3.1.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$9 x$	+	$10$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$x$
					+	$9 x^{2}$	+	$9 x$
							+	$10 x$	+	$10$
							+	$10 x$	+	$10$

Step 3.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $10 x + 10$

				$2 x^{2}$	-	$9 x$	+	$10$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$x$
					+	$9 x^{2}$	+	$9 x$
							+	$10 x$	+	$10$
							-	$10 x$	-	$10$

Step 3.1.1.5.15

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

				$2 x^{2}$	-	$9 x$	+	$10$
$x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$x$	+	$10$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$9 x^{2}$	+	$x$
					+	$9 x^{2}$	+	$9 x$
							+	$10 x$	+	$10$
							-	$10 x$	-	$10$
										$0$

Step 3.1.1.5.16

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

$2 x^{2} - 9 x + 10$

Step 3.1.1.6

Write $2 x^{3} - 7 x^{2} + x + 10$ as a set of factors.

$(x + 1) (2 x^{2} - 9 x + 10) = 0$

Step 3.1.2

Factor by grouping.

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Step 3.1.2.1

Factor by grouping.

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Step 3.1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 10 = 20$ and whose sum is $b = - 9$ .

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Step 3.1.2.1.1.1

Factor $- 9$ out of $- 9 x$ .

$(x + 1) (2 x^{2} - 9 x + 10) = 0$

Step 3.1.2.1.1.2

Rewrite $- 9$ as $- 4$ plus $- 5$

$(x + 1) (2 x^{2} + (- 4 - 5) x + 10) = 0$

Step 3.1.2.1.1.3

Apply the distributive property.

$(x + 1) (2 x^{2} - 4 x - 5 x + 10) = 0$

Step 3.1.2.1.2

Factor out the greatest common factor from each group.

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Step 3.1.2.1.2.1

Group the first two terms and the last two terms.

$(x + 1) ((2 x^{2} - 4 x) - 5 x + 10) = 0$

Step 3.1.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$(x + 1) (2 x (x - 2) - 5 (x - 2)) = 0$

Step 3.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$(x + 1) ((x - 2) (2 x - 5)) = 0$

Step 3.1.2.2

Remove unnecessary parentheses.

$(x + 1) (x - 2) (2 x - 5) = 0$

Step 3.2

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

$x + 1 = 0$

$x - 2 = 0$

$2 x - 5 = 0$

Step 3.3

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

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Step 3.3.1

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

$x + 1 = 0$

Step 3.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.4

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

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Step 3.4.1

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

$x - 2 = 0$

Step 3.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.5

Set $2 x - 5$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Step 3.5.2

Solve $2 x - 5 = 0$ for $x$ .

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Step 3.5.2.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 3.5.2.2

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 3.5.2.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 3.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 3.6

The final solution is all the values that make $(x + 1) (x - 2) (2 x - 5) = 0$ true.

$x = - 1, 2, \frac{5}{2}$

Step 4