Trigonometry Examples

$2 y^{4} + 3 y^{3} - 42 y^{2} + 68 y - 24 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$3 y^{3} - 24 + 2 y^{4} - 42 y^{2} + 68 y = 0$

Step 1.2

Factor $3$ out of $3 y^{3} - 24$ .

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Step 1.2.1

Factor $3$ out of $3 y^{3}$ .

$3 (y^{3}) - 24 + 2 y^{4} - 42 y^{2} + 68 y = 0$

Step 1.2.2

Factor $3$ out of $- 24$ .

$3 y^{3} + 3 \cdot - 8 + 2 y^{4} - 42 y^{2} + 68 y = 0$

Step 1.2.3

Factor $3$ out of $3 y^{3} + 3 \cdot - 8$ .

$3 (y^{3} - 8) + 2 y^{4} - 42 y^{2} + 68 y = 0$

Step 1.3

Rewrite $8$ as $2^{3}$ .

$3 (y^{3} - 2^{3}) + 2 y^{4} - 42 y^{2} + 68 y = 0$

Step 1.4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = 2$ .

$3 ((y - 2) (y^{2} + y \cdot 2 + 2^{2})) + 2 y^{4} - 42 y^{2} + 68 y = 0$

Step 1.5

Factor.

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Step 1.5.1

Simplify.

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Step 1.5.1.1

Move $2$ to the left of $y$ .

$3 ((y - 2) (y^{2} + 2 y + 2^{2})) + 2 y^{4} - 42 y^{2} + 68 y = 0$

Step 1.5.1.2

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

$3 ((y - 2) (y^{2} + 2 y + 4)) + 2 y^{4} - 42 y^{2} + 68 y = 0$

Step 1.5.2

Remove unnecessary parentheses.

$3 (y - 2) (y^{2} + 2 y + 4) + 2 y^{4} - 42 y^{2} + 68 y = 0$

Step 1.6

Factor $2 y$ out of $2 y^{4} - 42 y^{2} + 68 y$ .

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Step 1.6.1

Factor $2 y$ out of $2 y^{4}$ .

$3 (y - 2) (y^{2} + 2 y + 4) + 2 y (y^{3}) - 42 y^{2} + 68 y = 0$

Step 1.6.2

Factor $2 y$ out of $- 42 y^{2}$ .

$3 (y - 2) (y^{2} + 2 y + 4) + 2 y (y^{3}) + 2 y (- 21 y) + 68 y = 0$

Step 1.6.3

Factor $2 y$ out of $68 y$ .

$3 (y - 2) (y^{2} + 2 y + 4) + 2 y (y^{3}) + 2 y (- 21 y) + 2 y (34) = 0$

Step 1.6.4

Factor $2 y$ out of $2 y (y^{3}) + 2 y (- 21 y)$ .

$3 (y - 2) (y^{2} + 2 y + 4) + 2 y (y^{3} - 21 y) + 2 y (34) = 0$

Step 1.6.5

Factor $2 y$ out of $2 y (y^{3} - 21 y) + 2 y (34)$ .

$3 (y - 2) (y^{2} + 2 y + 4) + 2 y (y^{3} - 21 y + 34) = 0$

Step 1.7

Factor.

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Step 1.7.1

Factor $y^{3} - 21 y + 34$ using the rational roots test.

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Step 1.7.1.1

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.

$p = \pm 1, \pm 34, \pm 2, \pm 17$

$q = \pm 1$

Step 1.7.1.2

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

$\pm 1, \pm 34, \pm 2, \pm 17$

Step 1.7.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 1.7.1.3.1

Substitute $2$ into the polynomial.

$2^{3} - 21 \cdot 2 + 34$

Step 1.7.1.3.2

Raise $2$ to the power of $3$ .

$8 - 21 \cdot 2 + 34$

Step 1.7.1.3.3

Multiply $- 21$ by $2$ .

$8 - 42 + 34$

Step 1.7.1.3.4

Subtract $42$ from $8$ .

$- 34 + 34$

Step 1.7.1.3.5

Add $- 34$ and $34$ .

$0$

Step 1.7.1.4

Since $2$ is a known root, divide the polynomial by $y - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{y^{3} - 21 y + 34}{y - 2}$

Step 1.7.1.5

Divide $y^{3} - 21 y + 34$ by $y - 2$ .

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Step 1.7.1.5.1

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

$y$

$2$

$y^{3}$

$0 y^{2}$

$21 y$

$34$

Step 1.7.1.5.2

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

					$y^{2}$
	$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$

Step 1.7.1.5.3

Multiply the new quotient term by the divisor.

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			+	$y^{3}$	-	$2 y^{2}$

Step 1.7.1.5.4

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

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$

Step 1.7.1.5.5

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

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$

Step 1.7.1.5.6

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

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$21 y$

Step 1.7.1.5.7

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

				$y^{2}$	+	$2 y$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$21 y$

Step 1.7.1.5.8

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$2 y$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$21 y$
					+	$2 y^{2}$	-	$4 y$

Step 1.7.1.5.9

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

				$y^{2}$	+	$2 y$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$21 y$
					-	$2 y^{2}$	+	$4 y$

Step 1.7.1.5.10

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

				$y^{2}$	+	$2 y$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$21 y$
					-	$2 y^{2}$	+	$4 y$
							-	$17 y$

Step 1.7.1.5.11

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

				$y^{2}$	+	$2 y$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$21 y$
					-	$2 y^{2}$	+	$4 y$
							-	$17 y$	+	$34$

Step 1.7.1.5.12

Divide the highest order term in the dividend $- 17 y$ by the highest order term in divisor $y$ .

				$y^{2}$	+	$2 y$	-	$17$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$21 y$
					-	$2 y^{2}$	+	$4 y$
							-	$17 y$	+	$34$

Step 1.7.1.5.13

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$2 y$	-	$17$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$21 y$
					-	$2 y^{2}$	+	$4 y$
							-	$17 y$	+	$34$
							-	$17 y$	+	$34$

Step 1.7.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 17 y + 34$

				$y^{2}$	+	$2 y$	-	$17$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$21 y$
					-	$2 y^{2}$	+	$4 y$
							-	$17 y$	+	$34$
							+	$17 y$	-	$34$

Step 1.7.1.5.15

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

				$y^{2}$	+	$2 y$	-	$17$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$21 y$	+	$34$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$21 y$
					-	$2 y^{2}$	+	$4 y$
							-	$17 y$	+	$34$
							+	$17 y$	-	$34$
										$0$

Step 1.7.1.5.16

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

$y^{2} + 2 y - 17$

Step 1.7.1.6

Write $y^{3} - 21 y + 34$ as a set of factors.

$3 (y - 2) (y^{2} + 2 y + 4) + 2 y ((y - 2) (y^{2} + 2 y - 17)) = 0$

Step 1.7.2

Remove unnecessary parentheses.

$3 (y - 2) (y^{2} + 2 y + 4) + 2 y (y - 2) (y^{2} + 2 y - 17) = 0$

Step 1.8

Factor $y - 2$ out of $3 (y - 2) (y^{2} + 2 y + 4) + 2 y (y - 2) (y^{2} + 2 y - 17)$ .

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Step 1.8.1

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

$(y - 2) (3 (y^{2} + 2 y + 4)) + 2 y (y - 2) (y^{2} + 2 y - 17) = 0$

Step 1.8.2

Factor $y - 2$ out of $2 y (y - 2) (y^{2} + 2 y - 17)$ .

$(y - 2) (3 (y^{2} + 2 y + 4)) + (y - 2) (2 y (y^{2} + 2 y - 17)) = 0$

Step 1.8.3

Factor $y - 2$ out of $(y - 2) (3 (y^{2} + 2 y + 4)) + (y - 2) (2 y (y^{2} + 2 y - 17))$ .

$(y - 2) (3 (y^{2} + 2 y + 4) + 2 y (y^{2} + 2 y - 17)) = 0$

Step 1.9

Apply the distributive property.

$(y - 2) (3 y^{2} + 3 (2 y) + 3 \cdot 4 + 2 y (y^{2} + 2 y - 17)) = 0$

Step 1.10

Simplify.

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Step 1.10.1

Multiply $2$ by $3$ .

$(y - 2) (3 y^{2} + 6 y + 3 \cdot 4 + 2 y (y^{2} + 2 y - 17)) = 0$

Step 1.10.2

Multiply $3$ by $4$ .

$(y - 2) (3 y^{2} + 6 y + 12 + 2 y (y^{2} + 2 y - 17)) = 0$

Step 1.11

Apply the distributive property.

$(y - 2) (3 y^{2} + 6 y + 12 + 2 y \cdot y^{2} + 2 y (2 y) + 2 y \cdot - 17) = 0$

Step 1.12

Simplify.

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Step 1.12.1

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 1.12.1.1

Move $y^{2}$ .

$(y - 2) (3 y^{2} + 6 y + 12 + 2 (y^{2} y) + 2 y (2 y) + 2 y \cdot - 17) = 0$

Step 1.12.1.2

Multiply $y^{2}$ by $y$ .

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Step 1.12.1.2.1

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

$(y - 2) (3 y^{2} + 6 y + 12 + 2 (y^{2} y) + 2 y (2 y) + 2 y \cdot - 17) = 0$

Step 1.12.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(y - 2) (3 y^{2} + 6 y + 12 + 2 y^{2 + 1} + 2 y (2 y) + 2 y \cdot - 17) = 0$

Step 1.12.1.3

Add $2$ and $1$ .

$(y - 2) (3 y^{2} + 6 y + 12 + 2 y^{3} + 2 y (2 y) + 2 y \cdot - 17) = 0$

Step 1.12.2

Rewrite using the commutative property of multiplication.

$(y - 2) (3 y^{2} + 6 y + 12 + 2 y^{3} + 2 \cdot (2 y \cdot y) + 2 y \cdot - 17) = 0$

Step 1.12.3

Multiply $- 17$ by $2$ .

$(y - 2) (3 y^{2} + 6 y + 12 + 2 y^{3} + 2 \cdot (2 y \cdot y) - 34 y) = 0$

Step 1.13

Simplify each term.

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Step 1.13.1

Multiply $y$ by $y$ by adding the exponents.

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Step 1.13.1.1

Move $y$ .

$(y - 2) (3 y^{2} + 6 y + 12 + 2 y^{3} + 2 \cdot (2 (y \cdot y)) - 34 y) = 0$

Step 1.13.1.2

Multiply $y$ by $y$ .

$(y - 2) (3 y^{2} + 6 y + 12 + 2 y^{3} + 2 \cdot (2 y^{2}) - 34 y) = 0$

Step 1.13.2

Multiply $2$ by $2$ .

$(y - 2) (3 y^{2} + 6 y + 12 + 2 y^{3} + 4 y^{2} - 34 y) = 0$

Step 1.14

Add $3 y^{2}$ and $4 y^{2}$ .

$(y - 2) (7 y^{2} + 6 y + 12 + 2 y^{3} - 34 y) = 0$

Step 1.15

Subtract $34 y$ from $6 y$ .

$(y - 2) (7 y^{2} - 28 y + 12 + 2 y^{3}) = 0$

Step 1.16

Reorder terms.

$(y - 2) (2 y^{3} + 7 y^{2} - 28 y + 12) = 0$

Step 1.17

Factor.

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Step 1.17.1

Rewrite $2 y^{3} + 7 y^{2} - 28 y + 12$ in a factored form.

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Step 1.17.1.1

Factor $2 y^{3} + 7 y^{2} - 28 y + 12$ using the rational roots test.

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Step 1.17.1.1.1

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.

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

$q = \pm 1, \pm 2$

Step 1.17.1.1.2

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

$\pm 1, \pm 0.5, \pm 12, \pm 6, \pm 2, \pm 3, \pm 1.5, \pm 4$

Step 1.17.1.1.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

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Step 1.17.1.1.3.1

Substitute $0.5$ into the polynomial.

$2 \cdot {0.5}^{3} + 7 \cdot {0.5}^{2} - 28 \cdot 0.5 + 12$

Step 1.17.1.1.3.2

Raise $0.5$ to the power of $3$ .

$2 \cdot 0.125 + 7 \cdot {0.5}^{2} - 28 \cdot 0.5 + 12$

Step 1.17.1.1.3.3

Multiply $2$ by $0.125$ .

$0.25 + 7 \cdot {0.5}^{2} - 28 \cdot 0.5 + 12$

Step 1.17.1.1.3.4

Raise $0.5$ to the power of $2$ .

$0.25 + 7 \cdot 0.25 - 28 \cdot 0.5 + 12$

Step 1.17.1.1.3.5

Multiply $7$ by $0.25$ .

$0.25 + 1.75 - 28 \cdot 0.5 + 12$

Step 1.17.1.1.3.6

Add $0.25$ and $1.75$ .

$2 - 28 \cdot 0.5 + 12$

Step 1.17.1.1.3.7

Multiply $- 28$ by $0.5$ .

$2 - 14 + 12$

Step 1.17.1.1.3.8

Subtract $14$ from $2$ .

$- 12 + 12$

Step 1.17.1.1.3.9

Add $- 12$ and $12$ .

$0$

Step 1.17.1.1.4

Since $0.5$ is a known root, divide the polynomial by $2 y - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 y^{3} + 7 y^{2} - 28 y + 12}{2 y - 1}$

Step 1.17.1.1.5

Divide $2 y^{3} + 7 y^{2} - 28 y + 12$ by $2 y - 1$ .

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Step 1.17.1.1.5.1

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

$2 y$

$1$

$2 y^{3}$

$7 y^{2}$

$28 y$

$12$

Step 1.17.1.1.5.2

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

					$y^{2}$
	$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$

Step 1.17.1.1.5.3

Multiply the new quotient term by the divisor.

				$y^{2}$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			+	$2 y^{3}$	-	$y^{2}$

Step 1.17.1.1.5.4

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

				$y^{2}$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$

Step 1.17.1.1.5.5

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

				$y^{2}$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$

Step 1.17.1.1.5.6

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

				$y^{2}$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$	-	$28 y$

Step 1.17.1.1.5.7

Divide the highest order term in the dividend $8 y^{2}$ by the highest order term in divisor $2 y$ .

				$y^{2}$	+	$4 y$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$	-	$28 y$

Step 1.17.1.1.5.8

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$4 y$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$	-	$28 y$
					+	$8 y^{2}$	-	$4 y$

Step 1.17.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $8 y^{2} - 4 y$

				$y^{2}$	+	$4 y$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$	-	$28 y$
					-	$8 y^{2}$	+	$4 y$

Step 1.17.1.1.5.10

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

				$y^{2}$	+	$4 y$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$	-	$28 y$
					-	$8 y^{2}$	+	$4 y$
							-	$24 y$

Step 1.17.1.1.5.11

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

				$y^{2}$	+	$4 y$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$	-	$28 y$
					-	$8 y^{2}$	+	$4 y$
							-	$24 y$	+	$12$

Step 1.17.1.1.5.12

Divide the highest order term in the dividend $- 24 y$ by the highest order term in divisor $2 y$ .

				$y^{2}$	+	$4 y$	-	$12$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$	-	$28 y$
					-	$8 y^{2}$	+	$4 y$
							-	$24 y$	+	$12$

Step 1.17.1.1.5.13

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$4 y$	-	$12$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$	-	$28 y$
					-	$8 y^{2}$	+	$4 y$
							-	$24 y$	+	$12$
							-	$24 y$	+	$12$

Step 1.17.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 y + 12$

				$y^{2}$	+	$4 y$	-	$12$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$	-	$28 y$
					-	$8 y^{2}$	+	$4 y$
							-	$24 y$	+	$12$
							+	$24 y$	-	$12$

Step 1.17.1.1.5.15

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

				$y^{2}$	+	$4 y$	-	$12$
$2 y$	-	$1$		$2 y^{3}$	+	$7 y^{2}$	-	$28 y$	+	$12$
			-	$2 y^{3}$	+	$y^{2}$
					+	$8 y^{2}$	-	$28 y$
					-	$8 y^{2}$	+	$4 y$
							-	$24 y$	+	$12$
							+	$24 y$	-	$12$
										$0$

Step 1.17.1.1.5.16

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

$y^{2} + 4 y - 12$

Step 1.17.1.1.6

Write $2 y^{3} + 7 y^{2} - 28 y + 12$ as a set of factors.

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

Step 1.17.1.2

Factor $y^{2} + 4 y - 12$ using the AC method.

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Step 1.17.1.2.1

Factor $y^{2} + 4 y - 12$ using the AC method.

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Step 1.17.1.2.1.1

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 $- 12$ and whose sum is $4$ .

$- 2, 6$

Step 1.17.1.2.1.2

Write the factored form using these integers.

$(y - 2) ((2 y - 1) ((y - 2) (y + 6))) = 0$

Step 1.17.1.2.2

Remove unnecessary parentheses.

$(y - 2) ((2 y - 1) (y - 2) (y + 6)) = 0$

Step 1.17.2

Remove unnecessary parentheses.

$(y - 2) (2 y - 1) (y - 2) (y + 6) = 0$

Step 1.18

Combine exponents.

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Step 1.18.1

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

$(y - 2) (y - 2) (2 y - 1) (y + 6) = 0$

Step 1.18.2

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

$(y - 2) (y - 2) (2 y - 1) (y + 6) = 0$

Step 1.18.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(y - 2)}^{1 + 1} (2 y - 1) (y + 6) = 0$

Step 1.18.4

Add $1$ and $1$ .

${(y - 2)}^{2} (2 y - 1) (y + 6) = 0$

Step 2

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

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

$2 y - 1 = 0$

$y + 6 = 0$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

$y - 2 = 0$

Step 3.2.2

Add $2$ to both sides of the equation.

$y = 2$

Step 4

Set $2 y - 1$ equal to $0$ and solve for $y$ .

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Step 4.1

Set $2 y - 1$ equal to $0$ .

$2 y - 1 = 0$

Step 4.2

Solve $2 y - 1 = 0$ for $y$ .

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Step 4.2.1

Add $1$ to both sides of the equation.

$2 y = 1$

Step 4.2.2

Divide each term in $2 y = 1$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 y = 1$ by $2$ .

$\frac{2 y}{2} = \frac{1}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{1}{2}$

Step 4.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{2}$

Step 5

Set $y + 6$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $y + 6$ equal to $0$ .

$y + 6 = 0$

Step 5.2

Subtract $6$ from both sides of the equation.

$y = - 6$

Step 6

The final solution is all the values that make ${(y - 2)}^{2} (2 y - 1) (y + 6) = 0$ true.

$y = 2, \frac{1}{2}, - 6$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = 2, \frac{1}{2}, - 6$

Decimal Form:

$y = 2, 0.5, - 6$